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Electronics Research Laboratory

University of CaliforniaBerkeley 4, California

EXPERIMENTAL INVESTIGATION

OF MICROWAVE PROPAGATION IN A PLASMA WAVEGUIDE

by

R. N. Carlile

The research reported herein is made possible through supportreceived from the Departments of Army, Navy, and Air ForceOffice of Scientific Research under Grant AF-AFOSR-139-63.

August 12, 1963

TABLE OF CONTENTS

Page

ACKNOWLEDGMENT vi

SUMMARY vii

L INTRODUCTION 1

n. THEORETICAL BACKGROUND 5

A. The Permittivity Tensor 6

B. Lossless Plasma Waveguides 11

1. Surface Modes 12

2. Perturbed Waveguide Mode 15

C. Relation Between Collision Frequency and theAttenuation Constant 18

D. Plasma Waveguides With Loss 21

in. THE EXPERIMENTAL SYSTEMS 30

A. The Plasma Waveguide 30

B. The Traveling Wave System 31

1. Double-ring Coupler 31

2.. Attenuator 39

3. Rotating Coupler 39

4. Microwave Bridge 41

C. The Resonant System 42

1. Theory of Operation 42

2. Construction 44

D. Mercury-Vapor Discharge 44

1. Electrical Characteristics in the Presence

of a Magnetic Field 45

2. Tube Construction 48

-11-

Page

IV. SPECIAL TECHNIQUES 50

A. Measurements on Simultaneously Propagating Modes. . 50

B. Measurement of Plasma Frequency 51

1. Slater Perturbation Theory 51

2- E3q>eriment 53

3. Res\ilts 54

4. Error 58

V. RESULTS 61

A. The Backward-Wave Mode 61

B* The Perturbed TE^^^^ Waveguide Mode 79C. Noise and Drift *

VI. SUMMARY AND CONCLUSIONS 92

APPENDIX. ELECTRICAL CHARACTERISTICS OF A LOWPRESSURE DISCHARGE IN THE ABSENCE OF

A MAGNETIC FIELD 97

REFERENCES 101

LIST OF ILLUSTRATIONS

Figure 1: Cross-Section of the Plasma Waveguide 1

Figure 2: Phase Characteristics of the n = 0,1, 2 SurfaceModes on a Plasma Column Surrounded by a

Thin Dielectric Tube 14

Figure 3: Effect of the Permittivity of the Dielectric Tubeon the Cut-off Frequencies of the n = 1 Mode 16

Figure 4: Power Flow in a One-Dimensional System. 19

Figure 5: The Phase Characteristics of the n = 0 Mode atLarge Values of 2 | y | 26

Figure 6: The Attenuation Characteristics of the n = 0 Modeat Large Values of 21 y j 27

Figure 7: Mercury-Vapor Discharge Tube. Photograph ShowsTapered Aqua-Dag Attenuator 31

-111-

Page

Figure 8: Copper Waveguide and Probe Carriage 32

Figure 9: Axial Variation of Magnetic Field in Water-Cooled Magnet 34

Figure 10: Traveling Wave System 35

Figure 11: The Symmetric Double-Ring Coupler 37

Figure 12: VSWR of Symmetric Double-Ring Coupler 38

Figure 13: The 60° Non-Symmetric Double-Ring Coupler. ... 40

Figure 14: Microwave Bridge^ 41

Figure 15: The Resonant System 43

Figure 16: Plasma Frequency (fp) vs. Discharge Currentfor Tubes No. 2 and 3 55

Figure 17: Plasma Frequency (fp) vs. Discharge Currentfor Tube No. 3 with Magnetic Field as theParameter 56

Figure 18: ; Plasma Frequency (fp) vs. Magnetic Field forTube No. 3 with Discharge Current as theParameter 57

Figure 19: Plasma Frequency (fp) vs. Magnetic Field forTube No. 3 with Discnarge Current as theParameter 57

Figure 20: Brillouin Diagram for n = 0 and n = 1 Modesfor fp = 3. 0 Gc 62

Figure 21: Brillouin Diagram for n = 0 and n = 1 Modesfor f = 2. 7 Gc 63

P

Figure 22: Brillouin Diagram for n = 0 and n = 1 Modesfor fp = 2. 4 Gc 64

Figure 23: Brillouin Diagram for n = 0 Modes 65

Figure 24: Field Plot of a Quantity Proportional to vs.the Azimuthal Coordinate (j) for n = 1 Mode 67

Figure 25: Field Plot of a Quantity Proportional to Ej. vs.the Azimuthal Coordinate (j) for n = 0 Mode 68

-IV-

Page

Figure 26: Complex Propagation Constant y = P = ior vs.Frequency for the n = 1 Mode 69

Figure 27: Variation of Amplitude of n = 1 Mode vs.Distance Along the Waveguide 71

Figure 28: Complex Propagation Constant y = P = -orivs. Frequency in the Region of f^.^ 72

Figure 29a: Variation of n = 1 Cut-off Frequencies for aCase in Which fp Varies with Axial Position 78

Figure 29b: The Case where fp is a Constant with Respectto Axial Position 78

Figure 30: ^^11 Perturbed Waveguide ModeBrillouis Diagrams 81

Figure 31: Variation of Resonant Frequency of siiidTEi"]^2 Resonances with Magnetic Field 83

Figure 32: Signal to Noise Ratio for n = 0,1 Modes vs.Frequency; fp = 2. 4 Gc 86

Figure 33: Signal to Noise Ratio vs. Distance AlongPlasma Waveguide 87

Figure 34: Phasor Diagram of E^ at Five ObservationTimes 88

-V-

ACKNOWLEDGMENT

The author wishes to thank Professors T. E. Everhart, A. W.

Trivelpiece, and C. A. Desoer of this University and Professor

G. S. Kino of Stanford University for helpful discussions. Thanks are

also extended to Mr. G. Becker, who constructed the experimental

tubes; to Mr. J. Meade, who constructed the magnet; and to Mrs. Lynn

Schell, who prepared the manuscript; and, finally, to my wife, Marybeth,

for proofreading some of the material and for her interest and encourage

ment throughout this project.

The author would also like to acknowledge the loan of equipment

by the Department of Electrical Engineering of the University of Arizona.

The text of this report is similar to that of a thesis which has

been submitted to the Graduate Division of the University of California

at Berkeley, in partial fulfillment of the requirements for the degree of

Doctor of Philosophy.

-VI-

SUMMARY

The phase characteristics of a backward-wave pass-band mode

have been measured in a circular waveguide which is coaxial with a

circular isotropic plasma column. This mode has been identified as

an n = 1 surface wave similar to that described by Trivelpiece.

Zero temperature plasma waveguide theory, which predicts this

mode quite accurately, shows that a simple relation exists between the

lower cut-off frequency and the electron plasma frequency. It is demon

strated that the plasma frequency obtained from this relation, where the

cut-off frequency is determined from a measurement of the phase charac

teristics, agrees well with an independent measurement of the plasma

frequency using the well known cavity perturbation method.

Noise measurements on the n = 1 and the symmetric n = 0 modes

indicate that the signal-to-noise ratio is independent of input signal power

and is about 7 db for the n = 1 mode and varies from about 7 to greater

than 30 db for the n = 0 mode. A simple model of the plasma is proposed

which explains all the noise observations reported here. It is pointed out

that Crawford and Lawson independently propose a similar model.

A strong dc axial magnetic field was applied to the plasma and the

phase characteristics of the perturbed empty waveguide mode were

measured and are presented. In this case, the two degenerate circularly

polarized modes which constitute the mode in an empty waveguide

become non-degenerate and thus will have different phase characteristics.

The results are in qualitative agreement with the theoretical work of Bevc

and Everhart. It is demonstrated experimentally that as the magnetic

field is reduced, at a given frequency, the propagation constants of the

two modes approach each other and finally coincide at zero magnetic field.

-vii-

I. INTRODUCTION

A plasma waveguide is any longitudinally uniform transmission

system which, has as one of its constituents a plasma. From a struc-

/tural point of view, one of the simplest plasma waveguides is shown/

in Fig. 1. In this figure, the surface whose radius is c^ is metallic and

of high conductivity, i. e. , a copper surface. The region bounded by

radii c and b contains vacuum or air so that the dielectric constant € ^

in Fig. 1 is unity. The region bounded by radii b and a is a dielectric

tube with a scalar permittivity € i. e. , a glass tube, and the region

bounded by radius a is filled with plasma. Finally, this system is

immersed in a finite, longitudinal, constant magnetic field.

We see that this is a closed system with all fields contained with

in the metallic boundary. If c is of the order of an inch, we will not be

surprised to find that the frequencies of wave propagation occur in the

microwave range.

Fig. 1 Cross-Section of the Plasma Waveguide:Posiitive z-Axis Extends into Page.

The dielectric tube may in some cases be omitted so that instead

of the three-region system shown in Fig. 1, we would have a two-region

system of plasma and air. From a practical point of view, the function

of the dielectric tube is to contain the plasma and would be necessary if

the plasma, for example, was the positive column of a gaseous discharge.

On the other hand, it could and would be omitted if the plasma were a1-3

cesium thermal plasma which is contained by a magnetic field.

Due to its simplicity, this structure lends itself well to analysis.

In 1939 the two-region system was investigated by Hahn^ and Ramo^ inwhich the plasma was a drifting electron beam confined by a magnetic

field. In 1958, a thorough analysis of this system was made by Trivelpiece^in which a quasi-statis analysis was used throughout and the resulting modes

were slow modes and transverse magnetic (TM). This work was extended7

by Bevc and Everhart in which they carried the solutions into the fast-wave

regime. Various other workers have analyzed this system from specialized. ^ ^ . 8,-11

points of view.

The results of the analyses show that in the presence of a plasma the

empty waveguide modes become perturbed so that their propagation con

stants and field distributions are modified. Let us cause a non-symmetric

waveguide mode to propagate alternately as a left-hand and then as a right-

hand circularly polarized mode. If we introduce an isotropic plasma, we

would find that the oppositely rotating modes are degenerate in the sense

that both their propagation constants will be perturbed equally. If the plasma

is made anisotropic by the application of the magnetic field, then this degener

acy is removed and the two modes will have different propagation constants--

and therefore different--field distributions.

In addition to the perturbed waveguide modes, space charge waves. .6,7, 9-11 ^ ,

are tound. If the plasma is isotropic, an infinite set of space

charge modes will exist, the azimuthally symmetric mode being a low-pass

-2-

mode while the rest are passband modes. ^ These modes are called"surface waves" because their energy is concentrated at the plasma-

glass interface and also because there is no charge accumulation in

side the plasma, only on the plasma cylinder surface.

With the application of a magnetic field, additional low-pass

modes are introduced and a class of pass-band modes known as cyclo

tron modes ' appear. These latter modes are backward-wavemodes. All non-symmetric modes are non-degenerate.

Although the theory shows that an abundance of modes should

exist in this system, their experimental investigation has been some

what limited, and, particularly, there has been little good quantitative

agreement between theory and experiment. This project was started in

order to see what modes could be measured which had not been measured

previously, to develop new techniques, and to determine how well the

zero-temperature plasma waveguide theory will predict the modes that

are measured.

The symmetric low-pass space-charge mode with and without mag

netic field has been measured by Trivelpiece^ and also by Reidel andIZ

Thelan. They also were able to detect the presence of a backward-wave

cyclotron mode. Their work has been extended here to a measurement of

the non-symmetric mode whose field components vary as exp (i <j)). <j> is

the azimuthal coordinate shown in Fig. 1. The magnetic field for these

measurements is zero. This has been found to be a backward-wave pass-

band mode, and it is shown that zero-temperature plasma waveguide theory

predicts it and the accompanying symmetric mode quite precisely.

Historically, measurement of the perturbed waveguide modes has13

preceded space=charge mode measurements by at least a decade. While

a number of workers concerned themselves with perturbation by an

-3-

13-17isotropic plasma, Goldstein and his co-workers at the University

of Illinois and Pringle and Bradley have observed Faraday rota

tion of the ^waveguide mode perturbed by an anisotropic plasma.In the work reported here, the propagation constants of the two oppo

sitely rotating circularly polarized modes have been measured

for the case of a perturbing anisotropic plasma. In these measure

ments, a resonant cavity technique was used.

-4-

n. THEORETICAL BACKGROUND

We pre sent here a plasma waveguide theory which is pertinent to

the experimental results that have been obtained. In particular, we

consider surface waves for no magnetic field and perturbation of the

TEji waveguide mode. In both cases, loss is neglected. Finally, weconsider loss and its consequences.

The theoretical investigation of wave propagation in plasma wave

guides depends upon the solution of Maxwell's equations

VxE = - icou. HJ

VxH = ico€. E (2.1)

subject to appropriate boundary conditions. E and H are the vector

electric and magnetic fields and a time dependence of e^^^ for thesequantities is assumed, p. and € . are the permeability and permitti-

3 3

vity, respectively, of the j-th region of the waveguide. In all cases

considered here, p^~ scalar permeability of free space.If the j-th region is filled with plasma, then e . will in general

^ 22be a tensor. An expression for € . which is in wide use today is

where

€ . = €

3 o

v°

€ = € =1rr <[>((>

^rr ^r<j)

zzy

CO (1 - iv/co)P

2/1 . / i2 2CO (1 - iv/ CO ) -coc

-5-

(2.2)

(2. 3a)

r<j) (j)r

and 2CO ,

e = 1 - (2. 3c)

o) (1 - iv/u)

with

CO

2 ,CO CO / CO

, . = =i , P ^ (2.3b)(1 - IV / CO ) - CO

e = permittivity of free space io

CO = plasma angxilar frequency ,P

n = electron number density,e

e

CO = cyclotron angvilar frequency = B ,C* 111 o^ e

V = collision frequency,

m - electron mass,e

e = electron charge (absolute value).

Here, we have assumed the use of cylindrical coordinates r, <j), and z

shown in Fig. 1. Then E will have components E^, E^ and E^ withsimilar components for H.

A. THE PERMITTIVITY TENSOR

At this point, it will be worthwhile to consider the assumptions that

have been made in the expression for the plasma permittivity tensor

(Eq. 2. 2), where the plasma is the positive column of a mercury-vapor

discharge. This is the plasma which has been used in the experiments

described below. We may do this most easily by considering the Boltzmann

-6-

distribution function f(r, w, t) for electrons of the plasma. Each

electron of the plasma at a given time t may be specified by six

numbers, the three components of its spatial position vector r, and

the three components of its vector velocity w and so may be repre

sented by a point in a six-dimensional space called phase space which

has as its six coordinates the three coordinates of r and the three

coordinates of w. Let us call the three-dimensional space in which r

is the position vector, configuration space, and that in which w is the

position vector, velocity space. When all the electrons of the plasma

have been represented by points in phase space, f is the density of

these points in that space.

If no perturbing forces act on the electrons, an equilibrium dis

tribution will be reached which is invariant with time and may be repre

sented by Iw|) where we indicate that f^ depends on the magnitude of w and so is isotropic in w. If we now apply a small electric

field to the plasma, the electrons tend to move in the direction of the

field so that f is no longer isotropic in w, and may be represented by

an isotropic part Iw|) plus a small anisotropic part fj^(r,w,t). Ifthe electric field is suddenly removed, fj^ at every point in phase space

will decline to zero. We assume that we can represent this by

f^(r,w, t) ^ e' ^ (2.4)where we assume for the moment that v is a constant and has the

physical significance of being the relaxation frequency of the anisotropic

part of f. The mechanism by which the anisotropy in f is removed is

through encounters between the electrons and the other components of the

system, i. e., neutral atoms, ions, the container wall, and other electrons.23

V is commonly called the collision frequency and a rigorous analysis

shows that for the case of a billiard ball system which is the only case for

which the idea of collision frequency has a precise meaning, the expres

sion for V is just the classical collision frequency for that system.

-7-

From Eq. (2.4),

^ =-vfi. (2.5)

Now f = f + f- must obey Boltzmann's equation which in its mosto 1

general non-relativistic form is

+ w • V f + .V (2.6)8t m w \9t' ,,

e coll

where V is the gradient in configuration space and is the

gradient in velocity space. F(r,t) is the force on an electron at/fifN

r and t. I—) is the collision term. For a force free region^ 'coll _

and one in which there is no diffusion so that w. Vf is zero, we

have, using Eq. (2. 5),

=y

= - V f . (2.7)boll

We shall assume that Eq. (2. 7) is valid when external forces are

present and when the electrons experience diffusion, so that

Boltzmann's equation becomes

4^ +w . Vf + . V f =- vf • (2. 8)at m w 1

e

The average vector speed v of electrons at the point r in

configuration space is just the average of w at r over all possi-

ble velocities and weighted by f(r, w,t),

lwf(r,w, t) dwV (r,t) = 3 (2.9)

I f(r, w, t) dw ,

-8-

where dw is an element of volume in velocity space. Only con

tributes to the integral in the numerator. We recognize that J^f(r,w,t)dwis just n (r,t), the electron number density,

e

We now allow an rf electric field E to perturb the plasma and

allow a dc magnetic field B to exist. If we then multiply Eq. (2. 8)o

through by m w and integrate over all velocity space, it can be23 ®

shown that the following equation results:

0 _m {-^ +v.V)v=-e(E+vxB)-n Vp-n */m wvf dw

e 8t o e e U e L.-

(2.10)

where we assume that the pressure tensor / m ww dw can be represented

by pi, where I is the identity tensor and p is the scalar pressure, given

by an equation of state,

IpOCn

e

^ 10where % = constant.

Equation (2.10) is rather like the mathematical statement of

Newton's second law in which the left side is m dv/dt while the right

side consists of the applied forces. It is known as the equation of momen-23

tum transfer.

If the electric field is an rf field so that E = E(r) then we

may expect f to vary as e^^^ as will v, which is derived from f .— — -1Neglecting v . Vv and n^ Vp and assuming that v is a constant, Eq.

(2.10) becomes

(ico + v) V = - (E + V X B ) . (2. 11)m o

e

Using Eq. (2.11) and the relation for the rf current density

J = e n V ,e

-9-

it is a matter of simple algebra to obtain the expression for the per

mittivity tensor, Eqs. (2. 2) and (2. 3).

Although Eq. (2. 8) was derived in a rather non-rigorous fashion,23-25

it can be shown that the same result is obtained from a rigorous

approach. Equations (2.10) and (2.11) then result if v is independent of

w. It must also be independent of r or else the elements of the per

mittivity tensor will be a function of spatial position. Referring to Eq.

(2. 4), this means that the relaxation frequency must be the same at all22 - 24 ' .

points in phase space. It can be shown that for encounters

between electrons and atoms this will indeed be the case as long as the

electron doesn't approach the atom too closely. If encounters between

electrons and the wall are predominant, then we might expect the relaxa

tion frequency to be large at those values of w which cause electrons to

move to a wall quickly, i. e. , radially directed w, and small for those

values of w which tend to inhibit movement to the wall, i. e. , axially

directed w. Also, v may be a function of r.

We will point out in Table III. 1 that for the mercury-vapor dis

charge that has been used in the experiments to be reported in this work,

encounters between electrons and the wall are more than twice as great

as encounters between electrons and neutral atoms. Thus, conclusions

drawn from the assumption of constant v should be viewed with caution.

This is especially true when the magnetic field is zero. For a finite

magnetic field, wall encounters are reduced so that here, V may indeedbe a constant.

It should be clear that the effect of collisions on wave propagation

is to introduce loss. We have established that wave propagation depends

on an anisotropy in velocity being created by the electric field. Sincecollisions tend to destroy this anisotropy, they will consequently tend to

damp the wave.

-10-

Inclusion of the diffusion term n^ ^ Vp into the equation ofmomentum transfer leads to a small rf term in Eqs. (2.10) and (2.11)

, . , . icotwhich varies as e , but which has been calculated using the para-

_5meters of our system to be of the order of 10 smaller than the terme —

— E, and so can be neglected. The smallness of this term is due toe

the relatively low temperature of the electrons, and shows that tempera-

ature effects for our system are negligible.

Finally, we may summarize this Section by noting that the ex

pression for the permittivity tensor, Eqs. (2. 2) and (2. 3), should give

an accurate mathematical description of the plasma used in the experi

ments to be reported here, except that it may not be possible to repre

sent collisional effects by a scalar constant v.

B. LOSSLESS PLASMA WAVEGUIDES

Consider a plasma which is collisionless and therefore loseless.

We shall see how these results may be modified due to collisions in the

latter part of this section. For a collisionless plasma, v = 0 and the

elements of the permittivity tensor become simply,

2CO

p€ =€ =1--rr (b(b 2 2

CO - COc

CO (co /co)« , = =i —,— . (2.12)

rd) (br 2 2 ^ 'CO - CO

c

, 2, 2€ = 1 - CO /CO

zz p

We shall be interested in the solution of Eq. (2.1) for the case of zero

magnetic field (co^ = 0) which leads to surface waves, or waves whoseenergy is concentrated at the plasma-glass wall interface. Also, we

-11-

shall be interested in the perturbed waveguide mode.

1. Surface Modes = 0). This case has been treated in^ — c ^considerable detail by Trivelpiece, although he has not found all

the results that we shall need. Here, the permittivity of the plasma

is a simple scalar,

2€, /e = 1 - (o) /co)'

1 o p(2.13)

Trivelpiece finds that this case may be adequately treated by

using a quasi-static analysis in which only TM modes exist which are5 we

-ipz

2 2slow modes, i. e. j3 ^ ^ (oo/c) , where we assume that the z (axial)

dependence of all field components is e

Because a quasi-static analysis is used,

where

Then,

R (r) =n

E = -

, , ind) -iSz$ = R (r) e ^ e ^

n

(2.14)

A I (pr) inside the plasman

B[I (6r) K (Bb) - I (6b) K (pr)]outside the' ° " plasma

(2.15)

where I and K are the modified Bessel functions of order n, ann n

integer, and b is shown in Fig. 1. It is seen that we obtain an infinite

set of modes, one for each value of the integer n. Since the fields vary

as the modified Bessel functions, inside the plasma they will be largest

at the plasma edge, i. e. , at r = Outside the plasma, E^ and E^will decrease and become zero at the metal waveguide wall. Thus, the

energy stored in the electric field is concentrated in the region of the

plasma-glass interface. This fact becomes especially true when p is

large.

-12-

We may obtain a determinantal equation for p by taking the

solutions given by Eqs. (2.14) and (2.15) in the various regions and

applying the appropriate boundary conditions at the edges of a region.

In particular, we require that E and e E be continuous across

a boundary. The resulting determinantal equation is

2-

n. (j^) 1L \ " / J I„(Pa)

'2

(pb) - r(pb)K^(pa)] + Q[lJpb)K^(pa) - r(pa)K (pb)]

€2[ypa)K^ (Pb) - r (pb)K^(Pa)] + Q[ypb)KJpa) - ypa)K^{pb)]

(2.16)

whereI (Pc)KJpb) - r(pb)K (pc)

Q _ u n n nI (Pc)K (Pb) - I (pb)K (pc)

n n n n

for the three-region system shown in Fig. 1.

In Fig. 2, pa has been plotted versus co/w for n = 0, 1, 2CO

where

CO =00 / -t/I + €; (2.17)CO p ' 2 ^ '

It is seen that the symmetric mode (n = 0) is a low-pass mode, as has been

amply demonstrated theoretically and experimentally by Trivelpiece. ^ Thehigher-order modes, however, are seen to be backward-waves, each

occupying a narrow pass-band. By using the large argument approximations

for the various Bessel functions in Eq. (2-16), it is possible to determine

that all modes approach the common cut-off frequency co given by Eq.CO

(2.17) at large values of p.

If we consider a two-region system in which the plasma column of

radius a is bounded by a material of permittivity €^ which fills the space

-13-

oo

3

>OzUJ3oUJa:u.

1.4

1.2

ns2

n=|

f n=0

/

j0 > 3 4 5

1.0

0.8

0.6

0.4

0.2

PROPAGATION CONSTANT

Fig. 2 Phase Gharacteristics of the n = 0,1, 2 SurfaceModes on a Plasma Column Surrounded by a ThinDielectric Tube.

-lU-

between the plasma and the metal wall of radius b, then it has been

shown that the n = 1 mode is a forward mode (except at values of pa

lower than about 1/2) again occupying a narrow pass-band. The cut

off frequency at p = 0 is

CO = CO / V1 + p€ oup c-

(2.18)where

t.2 2p = ^ >1,P ^2 2

b - a

while the cut-off frequency at infinite p is again given by Eq. (2.17).

We can now discern that in the three-region system, the glass tube

causes the n = 1 mode (and probably the higher order modes) to be a

backward mode. Referring to Fig. 3, at p = 0, the cut-off frequency

CO i^^etermined by permittivity € which is an average of the permittivity of the glass tube and the air space weighted approximately

according to the volume occupied by each. This is because at p = 0,

the energy in the electric field is not concentrated near r = ^, and a

large portion of it will be found in the air space.

As p increases, the energy in the electric field becomes in

creasingly concentrated near r = ^ or in the glass. Thus, the effective

permittivity of the glass and air space approaches that of the glass so

that CO is continually decreasing. In the limit of infinite p, the ef-co

fective permittivity is just that of the glass, e This can cause

to be less than co with the result that we will have a backward-waveu

mode. This is illustrated in Fig. 3.

2. Perturbed TE^^^^ Waveguide Mode. If we consider a TE^^^ modein an empty circular waveguide, it is well known that this mode is linearly

polarized in the plane transverse to the waveguide axis. This linearly

-15-

CO

CO

(0u

COCO

^2 ^ ^2

Fig. 3 Effect of the Permittivity of the Dielectric Tube on theCut-off Frequencies of the n=l Mode.

polsirized rhodecsin be resolved into two circularly polarized modes

rotating in opposite directions but with the same propagation constant.

If we now introduce a plasma column on the waveguide axis and in

troduce an axial constant magnetic field may expect that the

resulting anisotropic plasma will have an unequal perturbing effect

on the two circiilarly polarized modes. This is because the electrons

of the plasma have a preferred sense of rotation and so will interact

-16-

with the mode which rotates in the same sense as they, in a different

manner from the mode which rotates in the opposite sense. We

designate the circularly polarized mode rotating in the same sense

as the electrons by + and the mode rotating in the opposite

sense by TE^^^" . Specifically, at a given frequency, we may expectP-- and therefore the fields of the two modes--to be different.

We might expect that if the radius of the plasma column was

small compared to that of the metal waveguide and if the plasma fre

quency were small compared to the signal frequency, the perturba

tions of p could be found by the perturbation techniques developed26 22

by Slater. Indeed, this has been done by Buchsbaum, et al. ,

in which they considered the perturbation of the resonant frequency

of a cavity resonant in the TE, mode. They found that the TE, ^Imn Imn

resonance shifts up in frequency from the TE resonance at co = 0,Imn c

while the resonance shifts down. The shift in frequency of

the resonance is larger than the frequency. Thus, we

would expect the plot of co vs. p for the TE^^^ mode to lie above theplot for the TE^^j^" mode on the Brillouin diagram.

These results are in accord with those of a formal field analysis7

of this case made by Bevc and Everhart. Like Trivelpiece, they

assumed that all field components have a variation proportional to

^i(a)t - pz + n(j)) (2.19)

but now we may expect that p for + |n| will be different from p for

- jn|. The variation of Eq. (2.19) indicates a field pattern which at

constant z rotates with an angular velocity of

<)> =- ^ , n 0 , (2. 20)

-17-

so that a mode characterized by a certain signed n is a circularly

polarized mode.

We shall not attempt to find the precise quantitative dependence of

to on p for the three-region system used in the experiments as was

done in part a above. The Slater perturbation technique is not useful

because of large to /to and because the plasma radius is not smallP

compared to waveguide radius. Referring to the results of Bevc and

Everhart, for a two-region system in which the plasma diameter is half

the waveguide diameter of 1. 20 in. (the same as in our experimental

system), for f = 1. 98 Gc. , they find that the cut-off frequency of the

TE ^ mode increases by 130 Mc. in going from f = Otof =2. 73 Gc.11 c c

For the case of a filled waveguide they show that the cut-off frequency

of the TE^j^~ mode changes much less than that of the mode. Atlarge values of p, the two modes coincide. Thus, in a measurement

of the TE^^^ and TE^j^" modes, we would expect to find on the Brillouindiagram the plot of the mode located approximately 130 Mc above

that for the TE^^" mode at small values of p, with the two merging atlarge p.

C. RELATION BETWEEN THE COLLISION FREQUENCY AND THEATTENUATION CONSTANT

In II. B it was assumed that v = 0 so that a mode propagates-ipz

without loss. Thus all field components propagate as e where p

is real. In the real world, v is not zero, losses are always present,-Wz t.

and we would expect a field component to propagate as e where

Y = p - ia is complex. We call p the propagation constant and or

the attenuation constant.

We would like to now show in a non-rigorous way the relation

between v and a. In the next section, we will show that approximately

the same expression can be obtained by using a rigorous approach.

-18-

P + dP

dZ

Z + dZ

Fig. 4 Power Flow in a One-Dimensional System.

We consider a one-dimensional system in which a single modepropagates in the + z direction as shown in Fig. 4. It is now convenient

to think of collisions between electrons and the other components of the

system as occurring in the billiard ball sense, i. e. , a collision takes

place in a time interval very much shorter than the time between colli

sions, before and after this collision time-interval there is no inter

action between an electron and the component of the system with which

it collides, and the collision is purely elastic. Thus, a collision is

well defined and so the collision frequency has a precise meaning. We

call this frequency v and assume it is a constant, independent of r

and w.

In Fig. 4 we represent the power per unit area of the mode at a

point z by P. If the plasma is lossy, the power per unit area will be

less at z + dz, namely, P + dP where dP is negative. The energy

-19-

stored in the mode will be partly stored in the electromagnetic fields

of the mode (in the electric field almost entirely if the wave is a slow

mode)and partly in the form of kinetic energy of the electrons.

There is a continual exchange of energy from the electromagnetic

form to the kinetic form. The electric field acts upon an electron, and

transfers mode energy to it. In doing this it gives it an rf velocity

which that electron must maintain in order to be able to give its mode

energy back to the electric field again. If the electron suffers a colli

sion, the direction of its rf velocity becomes changed and the electron

cannot give its mode energy back to the electric field. This energy

eventually becomes distributed among the other degrees of freedom of

the system and is lost to the mode.

In Fig. 4, let the number of electrons per unit volume be N and

call the velocity with which the energy of the mode travels, v , theO

group velocity. Since the energy per unit volume is P/v , we have thatO

the energy per unit area stored between z and z + dz is (P/v )dz, andO

that the energy carried per electron in dz is

20-

V N' g

where we introduce the factor — since on the average only total

stored energy is in the form of kinetic energy of the electrons.

Since an electron suffers v collisions per second, in one second1 P

one electron will cause - v units of energy to be lost and since^g

there are Ndz electrons per unit area in dz, the total energy per unit

area lost per second in dz is

dP = - ^ dz . (2. 21)2v

g

where the minus sign indicates power lost. Integrating Eq. (2. 21) yields

P = P e (2. 22)o

27But it is well known that power decays as

so that

P = P^ o (2. 23)

(2.24)

g

Thus we see that when a mode has its stored energy traveling

slowly the mode will propagate with greater loss than when the energy

velocity is high.

In the above, it would have been more correct to use energy

velocity than group velocity since in a medium which is dispersive,

28such as a plasma, the two are not always the same. Nevertheless,

we have asstimed their equivalence in this development.

D. PLASMA WAVEGUIDE WITH LOSS

We now present an introduction to the subject of plasma wave

guides with loss. We shall consider a specific case, namely the propa

gation of the n = 0 surface wave (co^ = 0) considered in II. B. We willuse only a two-region system (no glass tube) and will call the radius of

the plasma a as before but will call the radius of the metal waveguide

b. is the permittivity of the plasma and e^ is the permittivityof the material filling the space between the plasma and the metal wall,

the latter being a scalar constant.

For e we use Eqs. (2. 2) and (2. 3) where we will let ca =0.1 c

Then,

-21-

^1 '

^-1.U J ' 2 '

1

" 2 • 2 *

T

200

2- }L

1

1 + — (2. 25)CO

CO O) ^ ^ V

(0

We assume v is a constant of the system, and not a function of r

or w.

Anon-zero € ' will result in energy being absorbed from awave propagating in the system. We therefore assume an axial varia

tion of any field component of

where y = p - i or.

The calculation of the determinantal equation for y proceeds

exactly as shown by Trivelpiece and is identical to his Eq. (IV. 9)

except that his p has been replaced by our y and the term in

now appears. This equation is

€ ' 6 I^(y a) {y a) K^{y b) - I^(y b)K| (y a)^^ ~ iMy a) I^(y a)K^(Y b) - I^(y b)K^(Y a)

(2. 26)

The assumption by which this equation was derived is that

22-

|Re Y1 = - Qf^l >> |€ |

|lm Y1 =|2 Of p| >> h' 1 .(2. 27)

This and the assumption of a TM mode allows the quasi-static approxi

mation to be used where

E = - V ® •

The form of ® which will satisfy the differential equations is

$ = A (r) e"^"^ ^ .n

We shall be interested in the dependence of y on co, obtained

from Eq. (2. 26), in the vicinity of the large p cut-off frequency coCO

We may expect a to be fairly large here so that conditions stated in

Eq. (2. 27) should easily be met.

For large p, the Bessel functions in Eq. (2. 26) can be replaced

by the simple algebraic functions

'00 » I" > arg V>- ^ , (2. 28)

-Ve

VI—>00, ^ >arg V>- j

-23-

Using Eqs. (2. 28), Eq. (2. 26) becomes*

, €" - Y s Y s

+ i — = — — (2. 29)'2 '2

where s = b - a.

Note that for v = 0, y = p ^oo,

t

1 1

2u

1 E-€- V 2

(O

"p 6and CO = co = '— — , in agreement with Trivelpiece.

CO

+ ^2

Substituting p - i or for y in Eq. (2. 29) we find

' , -4pa -2ps .1 . 1 l-e'^+2ie'^ sin 2QfS

+ i — - ~ . n •

We now define

6_ . ~ -2ps -4ps2 2 1 - 2e cos 2ars + e

X=e"^P® , 1< X < 0 for P> 0, (2. 31a)

y = 2as, y > 0, (2. 31b)

tco^1 - "1 , A> 0for CO <(o / ^ /l +

CO

V2 '2

CO

(2. 31c)

This asymptotic form of the determinantal equation for y would befound for a surface wave of any order n using the asymptotic expressionsin Eqs. (2. 28). However the error in y , where y is obtained from it,is least for the n = 0 mode and increases rapidly with n, especially at lowvalues of y •

r.

-24-

Then

^ (>)(:) ;:iB= — (~"2 )(— I Z~j B > 0.

CO (2. 31d)

A =

B =

l-x^2 '

1 - 2x cos y + X

(2. 32)

X sin y2

1 - 2x cos y + X

The expression for A is divided into the expression for B to

give a quotient dependent on x and sin y. This may be solved for sin y.

The expression for A is solved for cos y. By using

2 2sin y + cos y = 1

an equation for x alone will result. This is a bi-quadratic in x, and it2

can be easily solved for x : '

^2 ^ (2B)^ ^ (A tl)^ . ^^ 33^• (2B) + (A + 1)

The negative solution must be discarded since x is positive or

zero. The solution with (A + 1) must be discarded since this would give a. . 2

trivial solution of x =1. We have, finally,

x2 = ^ . (E. 34)(2B) + (A + 1)

-25-

The expression for y may now be found:

4Bsin y = I 4 2 2 2 2\/(2B) +2(2B) (1 +A) + (A - 1)

When A = 1, w = CO . At this value of co,CO

-26sX = e =

, 1

sin y = sin 2as =

B^ + 1n/

CO = COCO

(2. 35)

(2. 36)

This value of x corresponds to af^proximately a maximum value

of p. Reference to Eq. (2. 34) shows that p will increase to this maximum

at CO = CO and then will decrease for co > co • This behavior is shownCO CO

in Fig. 5 for a set of typical parameters. Thus, p does not approach in

finity as in the lossless case, but has a maximum value determined by v.

^ CO / COp

n = 0

62=4.82V =.00E

TTp

CO

CO

p s•

plasma—

I

i

2ps^.1

Fig. 5 The Phase Characteristics of the n =0 Mode at Large Values of 2 |y |s.-26-

.1

Also we note from Eq. (2. 34) that p is unaffected by loss exce|St near

its maximum, i. e., very near (0 = 0) where sin 2as = 1 VCO

Thus, if

+ 1« 1.

TTZors < — , (2.37)

the parameter controlling p is co and not v. From the discussion inP

II. A we may then conclude that for the n = 0 mode, if Eq. (2. 37) is

satisfied, the permittivity given in Eq. (2.13) should allow an

accurate calculation of p.

y / on the other hand, is small at o) < co , increases to aboutCO

Tr/2 at CO = (0 , and then continues to increase to a maximum ofCO

sin y =4B IT, TT > y > _

- ^ 2(2B) +1 ^

for CO > CO . This behavior is shown in Fig. 6.CO ®

CO

cog.

-27-

n = 0

€_ = 4.82

JL =. 005

plasma

.4 .'8 i.2 1.6 Z'.O zTi ^8

Fig. 6 The Attenuation Characteristics of the n = 0 Mode at Large Values of 2|y1s.

It is now possible to obtain an expression for a in terms of

the group velocity subject to the limitations expressed in Eqs.(2. 27)

and (2.28). We further assume that A is enough different from unity2

so that the term (2B) in Eqs.(2. 34) and (2. 35) is negligible compared2 2

to (A - 1) or (A + 1) . This further limits the analysis to regions re

moved from where p has its maximum. Then Eqs. (2. 34) and (2. 35)

reduce to

A - 1X =

A + 1(2. 38)

sin 2 ars = —— . (2. 39)A - 1

-26s ,Now X = e so that

= .2 se-^P®= (2.40)dco dw V

g

where we recognize dco/dp as the group velocity, v . Differentiatingg

Eq. (2. 38) with respect to co, we have

^ =2 4^ ^—7 • <2.41)•1" (1 +Ar

The derivative of A with respect to o) may be obtained from Eq. (2. 31c),

^ = - i B. (2.42)du V

Equating Eqs. (2. 41) and (2. 40) and making use of Eq. (2. 42),

Eq. (2. 39) becomes,

sin 2 or s 1 v

2 s 2 Vg

-28-

(2.43)

For 2 or s < < 1, (co < co^q) we have

or = TT2 V

g

(2. 44)

Comparing this with Eq. (2. 24), we see that the simple derivation of

n. C yields a result which is qualitatively correct in that it shows the

correct dependence of a on the parameter of the system. It is quan

titatively correct to within a factor of 1/2.

J/

-29-

in. THE EXPERIMENTAL, SYSTEMS

The experimental systems used in this work were a traveling

wave system and a resonant system. The former was used to measure

the characteristics of the surface modes because their attenuation was

expected to be high over a large range of propagation constant; the latter

was used with the perturbed TE^^ mode where attenuation was expectedto be low. The pertinent structural details of these systems are des

cribed below.

A. THE PLASMA WAVEGUIDE

The basic unit in both systems is the plasma waveguide. This

consists of a copper circular waveguide 1. 2 in. in diameter coaxial with

a mercury-vapor discharge tube. The properties of this discharge are

described in HI. D of this section and in the Appendix. In all mea

surements reported here, the glass tube containing the discharge is

pyrex with an inner diameter of . 63 in. and an outer diameter of . 74 in.

The wall thickness of the glass was found to vary by + . 022 in- - The

discharge tube is shown in Fig. 7.

In all measurements, the radial component of the electric fi6ld,

E , in the air space between the glass tube and the copper waveguider

was sampled by means of a small probe inserted through the copper wall.

In order to move the probe longitudinally, the copper waveguide was

slotted along its entire length. A sliding carriage through which the

probe protruded was made such that it filled the slot at all positions.

This can be seen in Fig. 8. It was necessary to fill the slot in anticipa

tion of exciting modes with circumferentially directed wall currents.

Mechanically, the system was nearly optimiim. There was no play be

tween the carriage and the waveguide.

30-

ANODE

LIQUID MERCURY TRAP

The magnet used to provide the magnetic field in these measure

ments was 26 in. long with a 4 in. bore. The magnet current was time-

regulated to approximately one part in one-thousand. The magnetic axis

of symmetry (assumed to be a straight line) did not coincide with the

geometric axis of symmetry so that extraordinary measures had to be

taken to insure that the plasma waveguide was coaxial with the magnetic

axis. At large magnetic fields, it was found that on the magnet axis and

at the center of the magnet a region 7. 5 in. long existed at the extreme

ends of which the magnetic field had dropped off by 1 percent of its value

at the magnet center. For a region 13. 5 in. long, the field at each end

had dropped by 3 percent. The active region of the plasma waveguide

was always located within this 13.5 in. region, so that the maximum

variation of the magnetic field over it was 3 percent. At low magnetic

fields, such as those used in the surface mode measurements, this is no

longer true. Fig. 9 shows the field variation on the axis as a function of

distance along the magnet at low fields. Here we see that the variation

of field over the active length of the plasma waveguide never exceeds one

gauss.

B. TRAVELING WAVE SYSTEM

This system, used in measuring the characteristics of the surface

mode , makes use of an input coupler and a waveguide termination. It is

shown in Fig. 10.

1. Double-ring coupler. It has been shown in Section 11 that the

surface mode which propagate in the absence of a magnetic field have a

z-component of electric field, E^, which varies radially as the modifiedBessel functions. For example, inside the plasma, for a mode whose

azimuthal phase varies by 2mr around the circumference,

E =AI (pr) e"^^^ (3.1)zn n

-33-

1. 0-

0. z-

Water-Cooled Magnet

6. 9 gauss

4. 2 gaus

Center of

Magnet

Use Scale at

Left in Gauss

4 8 12 16 20 24

Distance Along Magnet Axis (cm)

Fig. 9 Axial Variation of Magnetic Field in Water-Cooled Magnet.

-34-

COPPER WAVEGUIDE

TERMINATIONPLASMA

MOVABLEPROBE

~!5

DOUBLE-RING COUPLER

ANODE

>•

In the plasma, E^. is a maximum at the plasma edge, i. e. , at the

interface between the plasma and the'glass discharge tube. In the

air region outside the glass tube, E is a maximum at the air-tube

interface and then declines to zero at the copper waveguide wall.

Evidently, the energy of the mode is concentrated in the region of

the glass tube.

This fact suggests that two metal rings whose planes are

parallel to one another and which fit tightly over the glass tube could

be effectively used to couple energy into a surface mode. The rings

are connected to the inner and outer conductors of a coaxial trans

mission line. The electric field between the two rings is parallel to

E of a surface mode, and is applied at a position where E^ is a

maximum. Since the electric field on the coupler must be roughly

matched to E of the surface mode, we have a mechanism for couplingz

energy from the coaxial transmission line into a surface mode.

This coupler is shown in Figs. 10 and 11. In Fig. 12, the VSWR

looking into the transmission line feeding the coupler is shown as a func

tion of frequency. Also shown on this plot are the corresponding propa

gation constants (p) of the n = 0 and n =1 surface modes that were excited

by this coupler. It is seen that when p is large the VSWR is small,

reaching a minimum of 1. 4. This may be understood by reference to

Eq. (3.1), which shows that as p increases the energy of the mode be

comes increasingly concentrated at the plasma-glass interface and in the

region of the coupler. For smaller values of p, the energy is less con

centrated in the region of the coupler so that it is less effective in coup

ling energy into the modes.

According to Fig. 12, this coupler which is geometrically sym

metric was able to excite an n = 1 mode. This is probably due to the

electric field not being symmetric. The electric field between the two

rings is probably greatest at the point at which the rings are fed from the

-36-

V

Fig. 11 The Symmetric Double-Ring Coupler.

15.0-

cVSWR aNOPLASMA

^VSWR

.4 .6 .8 I.

FREQUENCY f (6c)

Fig. 12 VSWR of Symmetric Double-Ring Coupler.

-38-

coaxial transmission line and declines to a minimum on the opposite

side of the rings. No measurements have been made to determine the

maximum and minimum values of the field. If <|) is the azimuthal co

ordinate, with <j) = 0 occurring at the point at which the rings are fed,

according to the above reasoning, the exciting electric field between

the rings would vary as

l + acos(j), a<l.

The first term, unity, will cause the n = 0 mode to be excited.

Because the magnetic field is nearly zero and the plasma is isotropic,

the coupler will excite both the n = 1 and the n = -1 modes equally.

These two modes combined will have an azimuthal variation of E of

cos <j). The second term, a cos <j), then, will cause the n = 1, -1 modes

to be excited. In addition to the n = 0 and n = 1 modes, this coupler

under certain conditions was observed to excite modes with higher n.

Besides the geometrically symmetric coupler, a nonsymmetric

coupler was used in some measurements. Here, a complete ring was

replaced by a 60° arc of a ring. This coupler is shown in Fig. 13. It

had stronger coupling to the n = 1 mode than the symmetric coupler.

Also, it would excite modes of higher n.

2. Attenuator. This consisted of a tapered, aqua-dag coating on

the glass approximately 4 in. long and located at the end of the trans

mission region. It may be seen in Figs. 7 and 10. Like the double-ring

coupler, it depends for success on the mode energy being concentrated

in the glass. A typical VSWR for this termination for the n = 0 mode was

1. 2.

3. Rotating coupler. In order to determine the azimuthal varia

tion of a mode, a coupler mount was made so that discharge tube and

double-ring coupler would rotate as a unit. The coupler was glued to the

glass tube so that there could be no relative motion between the coupler

*Hereafter called the n = 1 mode.

-39-

1

''f/ iBi' ^ !

Fig. 13 The 60 Non-Symmetric Double-Ring Coupler.

and the glass tube. This arrangement allowed the field pattern of the

mode to be rotated and by means of the detection probe of III. A, the

azimuthal properties of the mode could be determined.

4. Microwave bridge. A schematic drawing of the microwave

bridge used with this system is shown in Fig. 14. The usual precautions

of isolating the two arms of the bridge from one another were observed.

6DBcw

/I

Source

V

vVW^

^0D3

0 •

Detector

X_Q? ^NS^N

Plasma Waveguide

Calibrated

Line Stretche^

—m—6DB

—W/V

3DB

—vwv

Line Stretchers(2)

Fig. 14 Microwave Bridge

Calibrated

Attenuator

The calibrated phase shifter was a calibrated line stretcher with an

extremely low VSWR; the calibrated attenuator was a calibrated co

axial attenuator whose attenuation was derived from a waveguide below

cut-off and whose length was made variable. Thus the propagation

constant across the waveguide is zero and ideally the phase shift across

the attenuator is constant, independent of the attenuation setting. This

-41

behavior was confirmed experimentally. The detector that was used

with the bridge was either a receiver with high sensitivity or a spectrum

analyzer which allowed visual inspection of noise components present.

C. THE RESONANT SYSTEM,

This system was used to measure the characteristics of the

perturbed TE^^ waveguide mode. It has the advantage of being muchsimpler than the traveling wave system but may only be used when the

mode to be investigated is relatively lossless. It is shown in Fig. 15.

1. Theory of operation. By placing shorting planes across the

plasma waveguide a distance L apart, a resonant system is created

which will resonate whenever the p of the corresponding waveguide

mode equals mir/L, where m = 1, 2, 3 Alternately, at resonance,

if i is the distance between two adjacent nodes of any field component,

then

P = ir/i . 2)

Thus, it becomes possible to find p as a function of frequency for the

waveguide mode by constructing a resonant system, causing it to reson

ate at a su;Cficiently large number of frequencies by varying L and m,

and at each resonance, finding p from Eq. (3. 2). For the TE^^^ wave

guide mode, the standard notation for a resonant mode for which p=mTr/L

^^llm *We have pointed out in Section II that in the presence of an aniso-

tropic plasma, the plasma waveguide will support both the TE^^+ andcircularly polarized modes which have different propagation con

stants and therefore different field distributions. Now, it can be shown

that ji either of these circiilarly polarized modes encounters a conducting

wall across the plasma waveguide, a single reflected wave wUl result

which has the same propagation constant, rotates in the same absolute

-42-

CO

PP

ER

WA

VE

GU

IDE

MO

VA

BL

EP

RO

BE

FIX

ED

WA

LL

EX

CIT

AT

ION

PR

OB

E

Fig

.1

5.

Th

eR

eso

nan

tS

yst

em

.

sense, and has the same field configuration as the incident wave.

Thus, a resonance of this circularly polarized wave is possible, and

the resonant system method of determining the propagation constant

for the and modes in the presence of an anisotropic plasma

is useful.

2. Construction. The length L of the resonant system was

made variable by allowing one wall to be movable. This wall was a

cylinder which slid along the inside of the copper waveguide. The center

of the cylinder was bored out to a diameter which would just accommodate

the glass discharge tube and the tube was rigidly attached to the cylinder.

The plane of the anode coincided with the plane of the end of the cylinder

so that the conducting plane across the plasma waveguide was continuous

except for the ring-shaped area through which the glass tube passed. The

assembly of cylinder and discharge tube moved as a unit. This may be

seen in Fig. 15.

The shorting plane at the other end of the system was only a par

tial plane, and consisted of a metal ring which filled the space between

the glass tube and the copper waveguide. This is clearly imperfect. The

resonant system was excited by a small probe which coupled to the E^

field component, and the field distribution at resonance, particularly, the

location of nodes, was determined by the movable detection probe discussed

in III. A.

D. MERCURY-VAPOR DISCHARGE

A plasma is defined as a region containing equal numbers of ions

and electrons, and thus is electrically neutral. The plasma that has been

used exclusively in these experiments is the positive column of a dc mer

cury-vapor discharge. A typical discharge tube is shown in Fig. 7. The

cathode is indirectly heated and is either an oxide-cathode or an L-cathode*

of diameter . 600 in. The anode is a molybdenum disk of diameter . 600 in.

* Manufactured by S'emi-con Associates of California, Inc. , Watsonville,California, (trade name is Semicon Type S).

-44-

so that both anode and cathode have diameters that are nearly the

inner diameter of the glass tube. The separation between the anode

and the cathode is 13. 5 in. Typical values of parameters for this

discharge tube are given in Table 3.1. The characteristics of the

discharge for zero magnetic field are discussed in the Appendix.

1. Electrical characteristics in the presence of a magnetic field.

In the presence of a magnetic field, the motion of the charge particles

of the mercury-vapor discharge can be helical about the direction of the

magnetic field if the gyration radius is less than the diameter of the con

fining dielectric tube. For 20,000° K electrons, the critical magnetic

field at which the gyration radius is the same as the discharge tube radius

used in these experiments is 2. 8 gauss. For 500°K mercury ions, it is

245 gauss. In the presence of a strong magnetic field of 1000 gauss or

more, we may expect a charged particle to follow a magnetic field line

with a velocity equal to its thermal plus drift velocities in the axial direc

tion, while its position transverse to the field line is confined to the gyra

tion radius which will be small compared to the tube radius.

An observation which bears out this restricted type of motion was

made on an early tube which had a spiral oxide cathode. By accident,

the spiral turns became distorted so that there were two spots which had

very high emission. In the presence of a magnetic field of 1000 gauss

with the cathode inside the magnet far enough so that it did not experience

the fringing field, two bright lines (brighter than the surrounding glow)

parallel to the tube axis and the magnetic field were observed to originate

on these two spots. The intensity and sharpness of definition of these two

lines was quite well maintained along the entire length from cathode to

anode. We interpret this to mean that an electron leaving the cathode does

indeed follow a field line from its origin at the cathode to the anode. Fur

ther, because the lines did not lose definition, the electron suffers few

collisions.

Further evidence that the electrons follow field lines was available

when it was observed that if the cathode is located well outside the magnet

-45-

TABLE 3.1 - TYPICAL VALUES OF DISCHARGE TUBE PARAMETERS

QUANTITY

radius of discharge tube

mercury vapor pressure

discharge current

cathode-anode voltage

plasma frequency(electrons)

plasma frequency (ionS)

number density(electrons)

number density (ions)

number density(neutrals)

percent ionization

temperature (electrons)

rms thermal speed(electrons)

axial current density(electrons)

drift velocity(electrons)

mean free path(electrons)

electron-neutral

collision frequency(elastic)

electron-wall collision

frequency (elastic)

SYMBOL VALUE COMMENT

o

V.

pi

n

n.

n

n

w

. 36 in.

-31. 7 X 10 mmHg At room temperature T

of 75OF (240C)

. 5 amp

26 volts

2. 4 Gc.

4.1 Mc.

cathode-anode separation, 13. 5 in.

Tube No. 2. Determined

by cavity perturbationtechnique.

f ./f = -v/m /m/pi p ^ ^e 1

7. 4 X10^%m^ n =1. 24 x 10^^ f ^e p

7. 4 X10^%m^5540 X10^%m^

n. w n1 e

18n = 9. 68 X 10^ p/T

~ . 1 percent

26,000°K Ref. 29, p. 158 andRef. 30, p. 223.

1.1 XlO^cm/s V= 6. 74 x 10 -^Tt e

. 27 amp/cm j ~ I /anode areae o

2. 3 X 10 cm/s V, = j /end ''e e

4 cm.^

27 Mc.

68 Mc.)

-46-

Ref. 30, p. 30

V =

V = V /2aw t

so that the fringing field exists in the positive column, then the tube

glow contracts over the region of the fringing field to a cylinder of

small diameter on the tube axis. It retains this character over the

remaining length of the tube. Here it is hypothesized that the elec

trons are converged by the fringing field to the axis where they then

follow the field lines to-the anode.

It should be pointed out here that the "glow" of the discharge

tube is intimately associated with the electrons. It is the res\ilt of

direct excitation of normal atoms by electrons or of the collision of29

an electron and an atom in an excited state.

Assuming that electrons do follow field lines from the cathode

to the anode, subsequent tubes were constructed with a planar cathode

-whose diameter was only slightly less than the inner diameter of the

discharge tube. If this cathode was placed inside the magnet far enough

so that it does not experience the fringing field, and if it is uniformly

heated so that its emission is uniform over its surface, then we may

expect in the presence of a magnetic field that the electron density in

the positive col\imn will be relatively uniform across the cross-section

of the tube except near the dielectric wall where a sheath.with a

negative potential must exist.

The alteration of the discharge characteristics discussed in the

Appendix due to the magnetic field is far from clear. Certainly, the

magnetic field reduces wall currents of both electrons and ions. This

reduces electron-wall collisions and recombination. The latter results

in a reduction in ionization in the body of the positive column which in

turn causes the potential across the length of the positive column (or

electric field) to decrease. It was observed that, indeed, the potential

did drop across the tube by as much as 6 percent. The electron tempera-

ture also decreases.

-47-

2. Tube construction. Seven tubes (in some cases old tubes

were rebuilt) have been constructed for this project, although only

three were usable. All but the last had oxide cathodes and all of

these eventually suffered from deterioration of cathode emission.

The resulting temperature-limited cathode emission caused the

voltage drop across the tube to increase and to be increasingly sensi

tive to discharge current. This condition could be temporarily cured

by increasing the filament voltage. Several of the tubes had such short

lives that no data could be obtained from them.

It was found that the emission life of the cathode could be increased

by taking extra precautions to remove all residual gases from the tube

before it was finally sealed off. With the last oxide coated cathode tube

that was built, the tube was sealed off from the diffusion pump with an

ion-pump still attached. This pump was connected to the tube via a

cold-trap which excluded the mercury vapor from the ion-pump. This

pump was then allowed to continue to remove gases for about 24 hours

before it was sealed off. The result was a tube which lasted for approxi

mately 200 hours of use (Tube No. 2).

The final tube contained an L-cathode (Tube No. 3). It was

processed in the same way as the tube described above. It has

lasted for at least 300 hours of operation with no indication of cathode

emission deterioration.

The anode consists of a disc of molybdenum which has a highly

polished planar surface, with a diameter only slightly less than that of

the inner diameter of the discharge tube. It was observed that under

conditions of poor cathode emission, the anode would glow red at moder

ate discharge currents. By attaching a finned structure onto the anode

lead and blowing air onto it, the anode was kept below the "red hot" con

dition.

-48-

Under certain conditions, brightly glowing spherical regions

appeared in front of*the anode and which moved about the anode sur

face in a random manner. These are probably the anode spots des-31

cribed by Armstrong, Emeleus, and Neill. It is not clear how to

insure that these spots will not be present, but from the experience

here and the work of others it appears that they will not appear if the

anode fills the discharge tube, it is highly polished, and is not too hot.

The vapor pressure in the tube was controlled by causing all

the mercury in liquid form to be located in a "trap" distant enough

from the active part of the tube to be at room temperature. The trap

is shown in Fig. 7. The condensation of the mercury in the trap was

accomplished by immersing the trap in liquid nitrogen for a period of

several hours. Since the mercury vapor tends to diffuse toward the

coolest part of the tube, in this case the trap, one would expect that

the temperature of the trap would control the vapor pressure. In

truth, good quality pressure control was not achieved, probably due

to a very slow pumping speed of the trap.

-49-

IV. SPECIAL TECHNIQUES

A. MEASUREMENTS ON SIMULTANEOUSLY PROPAGATING MODES

In Fig. 2, we have seen that there is a frequency range which

is common to both the n = 0 and n = 1 modes. Referring to that figure,

this frequency range extends from to the maximiim frequency of

propagation of the n = 0 mode. In Section V, we will find that the double-

ring coupler will indeed excite both modes in this frequency range. Thus,

we need a method for measuring the propagation characteristics of the

two modes while they are propagating simultaneously.

If the complex propagation constants of the two modes are Y 2~

p - i a , then the real parts of the propagation constants, p and p ,1,21,.2

can easily be found from the interference pattern of the two modes. Let

A(z) be the complex voltage induced in the detection probe, which can

move in the z direction, by the radial component of the electric field of

the composite wave. Then,

[-or z -ip zE^e e + E^e e J (4.1)

where E., and E are the radial components of the electric field ofX M

the two modes which excite the probe at z = 0, and may be assumed real

and positive without loss of generality, k is a proportionality constant.

The magnitude of A(z) will be characterized by a series of

maxima and minima (the interference pattern). We note that z = 0

occurs at a maximum. The next maximum will occur a distance 2i away

where 2i is determined by the relation

-i2p,i -i2p f0=6^^ (4. 2)

so that if 0 is the shift in phase of A(z) in going from one maximum to

-50

the next,

0 = + Ztt (4. 3)

where we assume > ^2' Since,from Eq. (4. 3),

-ip^f ip^ie = - e ,

at z = i there will be a minimum. Thus, i is the separation of a

maximum and the adjacent minimum.

If we measure 0 by using the microwave bridge described in

Section III, and if i is known, then p^^ can be found from Eq. (4. 3),and p^ from

^2 " ^1 "

We note that the accuracy with which p and p can be deter-

mined depends on how precisely 0 and f can be measured, and

<*2 cannot be related by such simple relations to measureable quantities,and in general can be found only after four measurements on A(z) have

32been made. In Fig. 28 in Section V, the method described above was

used to determine the values of p for the n = 0 and n = 1 modes at those

frequencies where these two modes propagated simultaneously.

B. MEASUREMENT OF PLASMA FREQUENCY

1. Slater Perturbation Theory. As an independent method of

determining the plasma frequency of the electrons in the positive column

of the mercury-vapor discharge, a cavity perturbation technique was used.26

Following Slater, if a cavity resonant in one of its normal modes is

perturbed by partially filling its volume with a dielectric material of ten

sor permittivity e , then the quality factor Q and resonant frequencydi

0) of the empty cavity will be shifted to new values co and Q given by3i

-51-

/1\ _ -<o Jt\ 2Q/ " 20)^

/ ^ ( 1 \ • Au) _ iu[ 10^ J ' [zqJ ' 0)^ 2u^ r g.-

^ j ._ - .._(« -I)E-E,dv2Q ZQ^ \ I E-E^dv

(4.5)

Here, E and E are the vector electric fields in the unperturbed3l *

and perturbed cavity, respectively, t and v are the volumes of the

dielectric and cavity, respectively, and I is the identity operator.

If the perturbing dielectric does not perturb the electric field appre

ciably, so that the expansion of E in terms of the electric fields of

the normal modes consists of a predominant term, E , then

E * E dva a

/ (€ ' - I)E •E dvAco CO Jt a a

^a E^^ dv (4. 6b)a

where € = € ' + ie If the dielectric is a plasma region, then e is

given by Eqs. (2. 2) and (2. 3).

The mode in a cylindrical cavity is particularly advan

tageous because it has only one component of electric field, E^ , thecomponent parallel to the cavity axis. Thus, if the discharge tube is

placed coaxial with the cavity, with a magnetic field also parallel to

the axis, inside the plasma

E = a € Ea z zz z

where a is a unit vector in the axial direction. Finally, from Eq. (2. 2)z

and (2. 3),

-52-

where

(^= :r (-^) —CO

Aco

CO \co / 2

CO

M , (4. 7b)

I, , E_ dvM= j ^2 (4.7c)

E„, dv

so that •'M is a parameter dependent only on the unperturbed fields of2 2

the cavity. For v < < co , Eq. (4. 7b) shows that a simple relation

exists between the plasma frequency co arid the shift in resonant fre-P

quency of the mode due to the presence of the plasma. This

relation is valid whether or not a magnetic field is present. Note that

CO calculated in this manner is an average over the volume of the plasmaP 2

with a weighting function of Ez

2. Experiment. A copper cylindrical cavity was used which was

3. 690 in. in diameter and 5.115 in. long. The discharge tube was located

coaxially with the cavity by passing it through holes . 750 in. in diameter

in the top and bottom of the cavity. The glass tube was pyrex with an

inner diameter of . 63 in. and an outer diameter of . 74 in. The length

of the cavity was large compared to the diameter of these holes in order

to reduce the influence of end effects. With the tube in place, the cavity

was observed to resonate in the TM„^ mode at f =2. 321 Gc.010 a

The presence of the glass tube will distort the normal mode fields.

Thus a calculation of M using Eq. (4. 7c) is difficult. Alternately, M

may be found experimentally. In order to do this, a rod of Incite was

-53-

used which had a diameter of .633 in. , about that of the plasma. A

pyrex tube like that used in the discharge tube was placed in the cavity

with the Incite tube inside it, and the change in the resonant frequency

of the cavity was noted when the Incite rod was removed from inside

the tube. From Eq. (4. 6b),

(f - f^)/fM = - -

where 6 is the dielectric constant of Incite. An attempt was madeS.

to determine e^ by cavity perturbation techniques. To within theexperimental error, the value obtained agreed with that published in

33the literature, € =2. 58. The value obtained for M is . 051, sothat from Eq. (4. 7b),

f =4.4 Vf(f - f ) - (4. 8)p a

3. Results. Extensive measurements of plasma frequency

were made on Tubes Nos. 2 and 3 and are shown in Figs. 16-19.

In Fig. 16, it is seen that the square of the plasma frequency

is a linear function of the discharge current over a large range of

discharge current, which is in agreement with theory. The approxi

mately linear relationship between the square of the plasma frequency

and discharge current has been observed by other workers including16 15

Trivelpiece and Osborne, et al. Figures 18 and 19 show that at

small magnetic fields the plasma frequency increases rapidly with

magnetic field until about ICQ gauss is reached. This is probably as so

elated with a rapid decrease of wall currents and recombination at

small magnetic fields, and probably indicates that above 100 gauss

these processes change only slowly with magnetic field. This effect6

has been observed by Trivelpiece.

-54-

i cn

<ji

I

3.0

iu

Tu

be

#

12

.0

Tu

be

#3

10

.0

Tu

be

#2

Tu

be

#3

Di

$cha

.rge

Cu

rren

t(a

mp

s

Fig.

16Pl

asm

aFr

eque

ncy

(f)v

s.D

isch

arge

Cur

rent

for

Tube

sNo

.2

and

3.E

rro

rin

Diu

rnal

Var

iati

on,

+5%

,+

3%+

3%at

.3,

.6,

and

.9am

ps.

,R

espe

ctiv

ely;

End

Eff

ects

,1%

Exp

erim

enta

lPo

ints

Can

BeM

axim

umof

3%L

ow

.

I

3.

0

2.

0

10

00

gau

ss

1 20

00

Dis

ch

irg

eC

urr

en

t

Fig.

17Pl

asm

aFr

eque

ncy

(f)v

s.D

isch

arge

Cur

rent

for

Tube

No.

3w

ithM

agne

ticFi

eldas

th^P

aram

eter.

Erro

rinf

^:Di

urnal

Varia

tion,

+S%

++

3%at

.3,

.6,

and

.9am

ps.

,R

espe

ctiv

ely;

End

Eff

ects

,E

xper

imen

tal

Poi

nts

Can

Be

Max

imum

of3%

Low

.

3. a

o

>NoC<0

u*(Uu

ni

sCO

cciI—I

Ph

2. a

1. 0

Tube No. 3

Pyrex

. 30 Ampi5

400 1200 1600 2000

Fig. 18 Plasma Frequency (fp) vs Magnetic Field for Tube No. 3with Discharge Current as the Parameter. Error in fpiDiurnal Variation, + 5%, 3%at .3, .6 amps. , Respectively;End Effects, 1%, Experimental Points can be a Maximum of3% Low.

. 90 Amps

Tube No. 3

2.0

Axial Magnetic Field{Gauss)

Fig. 19 Plasma Frequency (fp) vs. Magnetic Field for Tube No. 3With Discharge Current as the Parameter. Error in fp!Diurnal Variation, + 5%, 3%, 3% at .3, .6, .9 amps. ,Respectively; End Effects, 1%; experimental Points can bea Maximum of 3% Low.

-57-

There was experimental evidence which showed that the values

of f for Tube No. 2 shown in Fig. 16 were larger than the trueP

values of f for some of the measurements in which this tube wasP

used. After this tube had been operated for well over 100 hours, the

cathode emission began to deteriorate. This was characterized by

a slight increase of voltage drop across the tube and increased sensi

tivity of voltage drop to current. Increasing the filament power would

improve but not entirely cure this condition. Measurements were con

tinued nevertheless, since the positive column of the discharge still

retained the characteristic qualities described in Section III and the

Appendix. Plasma frequency measurements reported in Fig. 16 were

made during the period of reduced cathode emission.

It was possible to estimate that f^ during the period of highemission was lower than for reduced emission by the percentages

shown in Table 4.1..

TABLE 4.1 FOR HIGH CATHODE EMISSION, fp FOR TUBE NO. 2(SHOWN IN FIG. 16). -SHOULD BE REDUCED BY THEINDICATED PERCENTAGES.

discharge current (amps. ) percent applies to

1. 0 7 Fig. 20

.8 7 Fig. 21

. 2 ~0

4. Error. The error in these experiments comes from three

sources. First, it should be noted that the perturbation method is in

herently an approximation. The approximation was introduced in Eq. (4. 6)

-58-

where we replaced E,the exact field in the presence of the plasma, by

Ej, , the field in the absence of the plasma. -22

However, Buchsbaum, et al. have indicated that the error introduced2

by this approximation will be small if (f /f) is not too large. Inciden-P

tally, they show that f determined by the perturbation method will be^ 2

less than the true f . Also, the error is greater, the greater is (f /f) .P P

An estimate of the error for our case can be made by comparing

/ 2(f /f) determined by the perturbation method with that obtained by anP

exact field analysis for the case where the glass tube is absent. By start-2

ing with a given Af/f, determined by the two methods can be found

with relative ease. For A f/f = . 0700, the perturbation method gives2 ,2

(f /f) of 1. 34 while the field equations give an (f /f) of 1. 42. AssiimingP P

that the field analysis yields the exact value of f /f, the perturbationP

method gives in this instance a value of f /f that is 2. 7 percent low. As22 ^ .

predicted by Buchsbaum et al. , the perturbation method results in a

conservative value of f /f. For the cavity used, an (f /f)^of 1. 34 corres-P P

ponds to an f of 2. 90 Gc. , roughly, the maximum f measured. ThusP P

2. 7 percent is about the maximum error expected from this source, which

we round off to 3 percent.

The second source of error is due to end effects. The length of

the cavity was made large compared to the glass tube diameter in order

to minimize this error. A hole was drilled in the end walls of the cavity

through which the glass tube passes. Certainly in the region of these holes

E is markedly different from so that the contribution of these regions

to the integrals in Eq. {4. 6) will cause error. We can minimize this error

by making the volume of these regions small compared to the total volume.

An estimate of the error introduced can be found by experimental

methods. For the Incite rod, e - 1 is 1. 58 which, it will be noted, has2

the same significance in the perturbation equation as an (f /f) of 1. 58.P

-59-

A f/f was measured for the rod and glass tube extending through the

wall in the manner in which all the measurements were made. Then

A f/f was measured for the case where one of the holes was filled with

a metal plug, and the glass tube and rod were thus terminated on a

continuous wall. For this latter case, E and E should be about theSL

same. A difference of about 1 percent was found between the two values

of Af/f, which leads to an error in f^/f of 1/2 percent. The error inf/f due to both ends of the cavity is twice this, or 1 percent.

PFinally, we recognize that there will be diurnal variations of

the electron concentration and of f for a given discharge currentP

probably due to diurnal variations in vapor pressure. We must then

ask, if f is measured on one day at a given discharge current, howP

much variation would we expect between this value and another made on

another day at the same discharge current?

An approximate answer to this question can be obtained by

comparing the results of four different series of measurements of fP

vs. discharge current scattered over a period of three months. This

comparison indicates that at nearly zero magnetic field, the mean varia

tion in f is less at large values of f than at small values. At dis-P P

charge currents of . 3, .6, and . 9 amp. , the mean variation is + 5 per

cent, .+ 3 percent, and + 3 percent, respectively. When a magnetic field

is present, this variation is somewhat larger.

-60-

V. RESULTS

The measurements were directed toward the investigation

of two plasma waveguide modes, a backward mode and a perturbed

waveguide mode, the results of which are presented below. In addition,

the results of measurements of noise and parameter drift in the plasma

are presented.

A. THE BACKWARD-WAVE MODE

In Section II we have seen that at no magnetic field the plasma

waveguide theory predicts that in addition to a low-pass forward wave

mode at no magnetic field there will be an infinite set of pass-band

modes. While the former is symmetric in the azimuthal coordinate <j)

(i. e. , n = 0), the azimuthal dependence of the latter are described by

n = 1, 2, 3,. .. . Since the fields of these modes vary as the modified

Bessel functions, their energy tends to be concentrated near the plasma-

dielectric interface, and hence these modes are called surface modes .

While the n = 0 mode is a forward mode, the high permittivity of the

dielectric causes the n = 1 mode to be a backward mode.

In this section, we describe an investigation of the n = 1 mode.

The traveling wave system described in Section III was used in which the

n = 1 mode was excited by a double-ring coupler.

In Figs. 20-23, Brillouin Diagrams of the n = 0 and 1 modes are

given with plasma frequency as the parameter. The magnetic field in

these measurements, and in fact in all measurements reported in this

section, except where noted, was the residual magnetic field in the

magnet which was measured to be 3. 5 gauss. It was not possible to

measure the n = 1 mode at f = 2. 0 or 1. 4 Gc. because of high attenua-IT

tion. High Attenuation also limited the range over which p cotild be

measured.

-61-

lli 0.8

2 3 4 5 6

PROPAGATION CONSTANT yS (In."')Fig. 20 Brillouin Diagram for n = 0 and n = 1 Modes for

fp = 3. 0Gc. (I^ =.98 amp. ). Error in p:Approximately + 3 Percent.

-62-

TUBE No. 2

fp=3.0 6c.(Iq= .98 a.)B^-3.5 Gauss

THEORY'

METAL

AIR

PYREX

PLASMA

0.74

Tube No ?

^ = 2. 7 GcBq = 3.5 Gauss(Iq = . 80 amp.)

Metal

0. 63" d am

0. 74" di; im

1. 20" dij im

Fig. 21 Brillouin Diagram for n = 0 and n =1 Modes for fp = 2. 7 Gc(Iq = . 80 amp. ). Error in p: Approximately + 3 Percent.

-63

I

I

Tube No. 2

f =2.4 Gc.P

Bq= 3. 5 Gauss(I = . 60 amp. )

Metal

Figc 22 Brillouin Diagram for n = 0 and n =1 Modes for fp = 2.4 Gc. (Iq - • 60 amp. ). Errorin P: Approximately 3 Percent.

MetalTube No. Z

B =3.5 gauss

0. 63" diam

0. 74"

1. 20"

1. 4

1.2

1. 0

o

O

oao

g-0)u

//////AI///

mn )

= 3.0 Gc

1. 3 Gc

Fig» 23 Brillouin Diagram for n = 0 Modes. Error in p:Approximately ^ 3 Percent.

-65-

We have pointed out in Section III that the azimuthal dependence

of E is proportional to cos ^ for the n = 1 mode. Since (the

component to which the detection probe is .sensitive) also has this de

pendence, we test this (|>-dependence by probing the mode azimuthally.

As explained in Section III, this was done by holding the probe fixed

and rotating both coupler and discharge tube. A radial plot of a quantity

proportional to E^ vs <}> is shown in Fig. 24 for the n =1mode. Althoughthis plot was made for a particular set of parameters, it is nevertheless

typical. Figure 25 shows a plot of the same quantities for the n = 0 mode.

Although both plots differ from the mathematically ideal (dashed

curves), it is clear that a cos ^ dependence is predominant in Fig. 24,

while Fig. 25 shows that E^ is independent of <|>. The distortion of theseplots probably is due to the glass wall being of non-uniform thickness. The

location of the glass discharge tube with respect to the copper waveguide

was checked and the length of the air gap between them was invariant to

. 002 in. On the other hand, the glass wall thickness varied by + . 002 in.

(measured in a similar glass tube). This thickness is critical since it is

in the glass that the energy is concentrated.

In addition to measurements of propagation constant, measurements

of a, the attenuation constant of the mode, were made. For the n = 1

mode typical plots of a are shown in Fig. 26. Also shown are the corres

ponding plots of p. It will be noted that the variation of a with frequency

is in qualitative agreement with Eq. (2. 44), which predicts that for a con

stant electron-collision frequency^ a will be inversely proportional to the

group velocity, dco/dp.

In Fig. 2 we have seen that the n = 2 surface mode occupies approxi

mately the same portion of the spectriim as the n = 1 mode. We may demon

strate that in the previous measurements, the system was relatively free

66-

27Cf

180

Tube Na 3

n = I

fp=2.4Gcf = 1206,

35 Gauss

Fig. 24 Field Plot of a Qtiantity Proportional to vs. theAzimuthal Coordinate <j> for n = 1 Mode.

-67-

270^

180'

0

Tube No. 3

n = 0

fp= 2.4 Gg

f = .90 G,

Bq=3.5 Gauss

Fig. 25 Field Plot of a Quantity Proportional to vs. theAzimutha'l Coordinate <i) for n = 0 Mode.

-68-

I nO I

1.

6

1.5

-

1.

4

1.

3-

1.

2

l.I

-

1.

0

o O Ix o a 0) g- 4) >-<

f=H

f=

3.1

Gc

£=

2.

8G

c

P(in

"^)

81

01

2is

.6

fp=

3.1

Gc

fp=

2.8

Gc

Hg

Pla

sm

aA

ir

Tu

be

No

.2

n=

1

B=

3.5

Gau

o

.7

Meta

l

1.

20

0"

.633

"'"1

h

.8

.9

Fig

.26

Co

mp

lex

Pro

pag

atio

nC

on

stan

ty

=P

=ia

vs.

Fre

qu

ency

for

the

n=

1M

ode.

Err

or

inp:

Ap

pro

x.

+3

Perc

en

t.

»s

from this and higher order modes by plotting relative amplitude vs.

distance at frequencies in^the n = 1 frequency band. This has been

done in Fig. 27 at one frequency;, the variation at that frequehcy

being typical of the variation at other frequencies in the n = 1 band.

Here it is seen that the maximum deviation from a purely linear varia

tion of amplitude with distance (the ^condition for total absence of higher

order modes) is very small.

It was of interest to determine the behavior of the n = 0 and n = 1

modes at large values of p where the excitation frequency is in the

neighborhood of the theoretical cut-off frequency (for a lossless system),

f =f / Vr+T. (5.1)CO p 2 ^ '

where c^ is the dielectric constant of the glass. In order to do this,

both p and or were measured near the coupler where the amplitude of

the mode was a maximum. This made measurements possible where a

is large.

The results are shown in Fig. 28. The interesting point to observe

here is that there exists a frequency band inside which both the n = 0 and

the n = 1 modes will propagate. From Fig. 28 we see that there was

propagation of both modes from 1. 06 to 1. 09 Gc. If attenuation had not

limited the measurements, we would have found that this band was more

extensive.

This behavior is predicted by the plasma waveguide theory developed

in II. B. Referring to Fig. 2, we see that the n = 0 mode has a maximum

frequency of propagation which is greater than f . Thus, there will in-co

deed exist a band of frequencies common to both modes which extends

from f^^ to the maximum frequency on propagation of the n = 0 mode.

-70

Distance Along Axis

(inj

Tube # 2

Bq = 3.5 Gauss

Fig. 27 Variation of Amplitude of n = 1 Mode vs. Distance Along theWaveguide. The Position Indicated by 1. 0 on the Abscissa isNear the Termination.

-71-

I

o

CD

1.2-

1.0.-

'0.8-'

>•ozUJz>oUJ 1.2oru.

I.e..

0.8"

ATTENUATION CONSTANT a (In.-')0.2 0.4 0.6 0.8 1.0 1.2 1.4

—I 1 1 1 1 + -h

4 5 6 7

PROPAGATION CONSTANT yS

TUBE No. 2

fp= 2.5 6c.3.5 Gauss

8

(In.-')Fig. 28 Complex Propagation Constant y = P = -ffi vs. Frequency in the Region of

Error May Be Large for All Quantities since Measurements were Made in aRegion Where fp was Varying with Axial Position (Region S in Fig. 29a).

PLASMA

PYREX

Asa final note, in measuring the n = 0 mode in Fig. 28, it was

not possible to make measurements at values of p greater than 6. 5(in.)

because the loss was too great. This is in accordance with Eq. (2. 44),

since for large values of p, da>/dp is very small.

We note that again Eq. (2. 44) gives the correct qualitative depen

dence of a on the group velocity assuming a constant electron-collision

frequency. Quantitatively, using Eq. (2. 44) both Figs. 26 and 28 predict

an electron-collision frequency of approximately 200 Mc. , which is in

rough agreement with the collision frequency predicted by cavity pertur

bation techniques and also Table 3.1.

In the measurements that have been discussed above, the backward

nature of the n = 1 mode was extremely evident. It was reported in Sec

tion III that continuous monitoring of the phase of a mode was possible by

means of a calibrated line stretcher in the reference arm of a microwave

bridge (see Fig. 14). For the n = 0 mode, as the detection probe was

moved away from the coupler the line stretcher had to be extended in

order to keep the bridge in balance, thus indicating a wave whose phase

was lagging as it proceeded away from the coupler. This is of course

the characteristic of a forward wave.

At a frequency in the n = 1 pass-band, as the longitudinal probe

moved away from the coupler, the line stretcher had to be shortened,

thus indicating a wave whose phase advanced as it proceeded away from

the coupler. This is the characteristic of a backward wave in which theI

phase velocity is toward the coupler while the*energy velocity must be

away from the coupler (the energy velocity must be away from the coup

ler for any wave).

We would now like to determine how well the theory developed in

II. B predicts the experimental results, in particular, the curves for the

n = 0 and n = 1 modes shown in Fig. 20.

-73-

The theoretical results presented in Fig. 2 have been obtained

for the dimensions of the experimental plasma waveguide. Thus, these

results may be directly compared with the experimental results in

Fig. 20 if the parameter f^^, the cut-off frequency, is available. Thiscannot be determined directly from Fig. 20 because it was not possible

(due to high attenuation) to make measurements at large enough values

of p to determine to what frequency the two curves are becoming asymp

totic. Therefore, the following procedure was adopted: f^^ was chosenso that a theoretical curve would coincide exactly with the corresponding

experimental curve at p=4. 5(in. ) ^.From Fig. 2, at p=4. 5(in.)' (pa =1. 42), f/f^^ =.99 for the n=0

-inode and f/f = 1.16 for the n = 1 mode. Now referring to Fig. 20, at

p = 4. 5 (in.) , f = 1. 24 Gc. for the n = 0 mode and f = 1. 44 Gc. for the

n = 1 mode. Thus, the theoretical n = 0 curve will coincide with the ex

perimental n = 0 curve at p = 4. 5 (in.) if f =1. 25 Gc. , while theCO ^

theoretical and experimental n = 1 curves will coincide at p = 4. 5(in.)

if f =1. 24 Gc. We split the difference and letCO

f = 1. 245 Gc.CO

The resulting theoretical curves for the n = 0 and n = 1 modes are

shown as dashed curves in Fig. 20. We note the close agreement nearly

everywhere between theory and experiment. In fact, we confirm that the

value of P chosen to make a theoretical and experimental curve coincide

for the purpose of determining f^^ is entirely arbitrary. Any value ofp could have been chosen yielding nearly the same value of f^^, and consequently the same excellent agreement between theory and experiment

wovild be obtained.

Referring to the experimental results shown in Figs. 21 and 22, once

again it can be demonstrated that there is close agreement between theory

-74-

and experiment.

From Eq. (5.1) we note that there is a definite relationship be

tween f and the plasma frequency f . For f = 1. 245 Gc. , theCO p CO

corresponding plasma frequency is f = 3. 00 Gc. where we have takenIt

€ =4. 82, the permittivity of pyrex. We may obtain an independent

check on this value of f^ by using the results of the cavity measurements reported in IV. B. In Fig. 20, the discharge current is . 99 amp.

Referring to Fig. 16, and taking into account the reduction factor in

Table 4.1, we see that the plasma frequency predicted by the cavity per

turbation method for this discharge current is f = 3. 25 Gc. , which isir

8 percent higher than that obtained above from Eq. (5.1).

It is of considerable interest to make a large number of compari

sons of f determined from Eq. (5.1) with f determined by the cavityp P

perturbation technique. From the data that has been taken on Tubes

Nos. 2 and 3, it has been possible to make nine comparisons. From a

plot of the n = 0 or the n = 1 mode at a certain discharge current, f^^was chosen so that the appropriate theoretical curve would coincide with

the experimental curve at (3 = 4. 5 (in.) . Then f was found from

Eq. (5.1). For the same discharge current, f was found from Fig. 16P

using Table 4.1.

The nine comparisons showed that the values of f found in the

two different ways differed on the average by 6. 2 percent. The largest

deviation was 11 percent and the smallest was 2 percent. In all cases,

the cavity-determined f^ was larger than the fp found from Eq. (5.1).This difference probably exists because the measurements used to obtain

the two values of f to be compared were made of the order of weeksP

apart. In terms of the discussion of errors at the end of IV. B, this

error can be attributed to diurnal variations.

-75-

The cavity-determined fp is an average of fp over the crosssection of the discharge tube with a weighting function that is nearly

unitv. f , however, depends on f^ at the discharge tube wall, since' CO P

at large values of p this is where the energy of the mode is concen

trated. We will point out in the Appendix that at the wall, the electron

density, and hence f^, may be of the order of 20 percent less than its£r

value on the discharge tube axis. We may therefore be tempted to

attribute the 6. 2 difference between the cavity-determined fp and fpobtained from f^Q to the decrease of the electron density at the wall.

This is probably not a valid conclusion. Trivelpiece^ has analyzeda case in which the electron number density n^ in the plasma has aparabolic radial dependence,

,, , r 2n = n (1 - (r)

e ea a

where n is the number density on the axis. The Brillouin diagramea

has been found for the n = 0 mode. Since the parameters for the case

that Trivelpiece analyzes are nearly identical to the parameters of the

experimental system used in this project (i. e. , nearly the same ratio

of inner to outer diameter of the discharge tube, nearly the same ratio

of discharge tube inner diaineter to Waveguide diameter, nearly the

same permittivity of glass), the results he finds should be applicable.

He finds that for p =. 2, or a 20 percent decrease in number density at

the plasma-glass tube interface, at p=5(in. ) ^( about the largest pfor the n = O.mode that was measured), the corresponding angular fre

quency CO is less than 1 percent lower than co for p ~ 0, or constant

number density across the cross section of the tube. Therefore, in the

range of p in^which the measurements reported here were made, the

Brillouin diagram for the n = 0 mode does not appear to be a sensitive

function of p .

-76-

In Fig. 26 it will be noted that the range of p in the n = 1 pass-

band extends from 4. 5 (in.) ^ to 11 (in.) ^while in Figs. 21 through 23,the range of p extends from 2(in.) ^ to about 7(in.) . By monitoringphase with distance it was possible to discern that in the former case

f was decreasing in the region of the coupler, as shown qualitatively inP

Fig. 29a, while in the latter case it remained fairly constant along the

tube, as shown in Fig. 29b. In the former case the emission was reduced

due to deterioration of cathode emission, while in the latter case the

emission was high. In the former case the anode was probably quite hot.

Since for the case of varying f , both the upper and lower cut-offP

frequencies of the n = 1 mode will vary in proportion to f ,then excita-P

tion of the n = 1 mode at the position of the coupler will result in propaga

tion within a restricted frequency range f^ ' to f ' down the length of thesystem. The measurements shown in Fig. 26 were made at the terminated

end of the system and so would be subject to this restricted range.

A wave at frequency propagates over the region S at a propa

gation constant away from cut-off and so experieyices relatively little

attenuation over this distance. Only as the wave front travels beyond this

range does it experience heavy attenuation. Thus, at the measuring posi

tion a wave at f 'in Fie. 29 a will have experienced the s^-me attenuationCO

as f 'in Fie. 26b where the former wave is nearer cut-off at the posi-co

tion of measurement and so will have a larger p than the latter wave.

We assume that at frequencies lower than two cases the atten

uation is so great at the measuring position that p cannot be measured.

A similar argument would show that if f^' are the frequencies in the twocases above which the attenuation is so great at the measuring position

that P cannot be measured, that at f ', p for the cast in Fig. 29a is^ u

larger than the p for Fig. 29b.

-77-

u

o

crou

^Z4

I Region ofMeasurement

Distance Along WaveguideTo Termination To Anode-

Fig. 29a Variation of n = 1 Cut-off Frequencies for a Case inWhich fp Varies with Axial Position.

>>u

o

u*o

Region of jMeasurement

••u

fCO

f u

CO

Distance Along Waveguide

-To Termination To Anode-

0)r—<

o

U

03

o

a,

Fig. 29b The Case where f is a Constant with Respect toJr

Axial Position.

-78-

Finally, the effect of magnetic field on the n = 1 mode was in

vestigated. Here, a 60° coupler was used to excite the system. It

was found that the only modes present to any extent up to about 8

gauss were the n = 1 and the n = 0 modes. No variation of propaga

tion constant or attenuation constant could be measured over this

range of magnetic field. Above 8 gauss additional modes were ex

cited in the pass-band of the n = 1 mode. Drift in the system did not

allow measurement of these modes by the techniques of IV. A. The

attenuation of all modes except the n = 0 mode increased rapidly with

magnetic field so that at 60 gauss the total power detected by the de

tection probe at a fixed longitudinal position had dropped by 30 db and

continued to drop at larger magnetic fields. There was no indication

that the frequency band of the n = 1 mode had shifted. It was observed

that the VSWR (about 6. 5) of the coupler did not change while the

magnetic field was being varied so that the power being coupled into

the modes remained constant.

B. PERTURBED TE^j^ WAVEGUIDE MODEThe resonant system technique described in III was used

to determine the variation of propagation constant p with frequency

for the perturbed TE^^j^ waveguide mode. The application of a dcmagnetic field causes the plasma column to become anisotropic so

that according to the discussion of II. C we may expect that the TE^j^+circularly polarized mode will have a different propagation constant

from the TE^^^- mode. In the resonant system there will be resonancesassociated with each of these modes which we have labeled TE^^ Im"^ and

TE ~ Here, m has the usual significance of being the number of1 Im ~

maxima of E on the axis. If i is the separation of adjacent minima,r

then

P = ir/f (5. 3)

•79-

This method of measuring p has proved to be useful and has

allowed the desired variation of p with frequency to be found for

the and TE^^^- modes. It is shown in Fig. 30.Generally, the Q of the resonant system was low, of the

order of 50. Nevertheless, the power ratio between successive

maxima and minima as measured by the detection probe was large,

typically, 20 to 30 db so that precise measurements of i , the dis

tance between minima, was always possible. In all determinations of

p, i was the distance from the anode-end wall to the first minimum.

Here it was assumed that the anode-end wall acts like a continuous

shorting plane across the system. The validity of this assumption

was confirmed by noting that £. determined in this way was indistin

guishable from the distance from the first minimum to the second.

It was found that sufficient points to determine the desired curves

could be obtained by varying the length of the resonant system from a

maximum of 6. 56 in. to about 4 in. with m taking on the values 2, 3, 4,

5. In Fig. 30, each point is labeled with the appropriate m. The points

for which m is circled were obtained with the resonant system at its

maximum length. It will be noted that the circled-m points which have

the same m do not have exactly the same p. The explanation for this

is two-fold. Firstly, since the "shorting-plane" at the cathode-end of

the system is merely a metal ring filling the gap between the glass tube

and the copper waveguide, the effective position of the reflecting plane

at this end may vary slightly with frequency. Secondly, the resonance

Q was low enough so that it was difficult to adjust the system to exactly

the condition of resonance. Indeed, in many (if not all) of the measure

ments, the resonant frequency was never exactly attained. This variance

from the resonant condition would account for the aforementioned points

not occurring at the same p. It should be pointed out here, that the

-80-

O 6.0

TUBE No. 3

fp.= 2.4 6c.

fc= 2.7 Go.

PLASMA

THEORETICAL

UNLOADED

WAVEGUIDE

META

AIR

PYREX

NO PLASMA

VELOCITY

OF LIGHT

I 2PROPAGATION CONSTANT p (In."')

+ ~Fig. 30 and perturbed Waveguide Mode

Brillouin Diagrams. The Number Indicates theType of Resonance Used to Obtain the Point,i. e. , 2 means TE^^Resonance. A Circle Indicates that the Cavity was at its Maximum Length.

-81-

exact condition of resonance is not at all essential in these measure

ments. The condition of resonance is significant only in that it affords

mode discrimination and allows effective coupling between a rather

simple coupler (a simple probe coupling to E^) and the resonant mode.The condition for accuracy of the measurements is that the ratio of a

field maximum to a minimum be large, indicating that only one mode is

present, and allowing a half-wavelength of the corresponding waveguide

mode to be accurately measured.

The variation of (3 with frequency shown in Fig. 30 is in agreement

with the qualitative discussion given in II. The curve labeled "no

plasma" was obtained with the waveguide loaded with the glass tube. The

dielectric loading will cause the stored energy per unit length to increase.26

Slater shows that for constant |3, this condition will cause the frequency

to decrease. Thus, the "no plasma" curve lies below the unloaded wave

guide curve in Fig. 30.

The introduction of plasma into the glass tube reduces the stored

energy per unit length by virtue of the negative dielectric constant of the

plasma. Thus, the TE^^^" curve which is little affected by the magneticfield (Fig. 31) will lie above the "no plasma" curve. Finally, we have

seen in II that the TE^^^"^ curve will lie above the TE^^" curve by approximately 130 Mc. for f^ = 2. 73 Gc. This is roughly in agreement with Fig. 30.

Figure 31 shows the dependence of the TEj^j^+ and TE^^^" modes on themagnetic field. Specifically, the *^^112" with theresonant system at its maximum length are shown. Careful checking shows

that any TE , + resonance will have the same dependence as does the T,.-+11m 11^

shown in Fig. 31, Any resonance will have the same dependence as

the TE^j^2" resonance.It is seen that the TE resonance is insensitive to magnetic field

except at low fields while the TE^^^"*" re sonants is quite sensitive to magneticfield. Below 500 guass it was not possible to detect the latter resonance

-82-

I 00

U)

1

5.

80

--

0)5

.7

0

Tu

be

No

.3

f=

2.

4G

cP

1.

20 0.

633"

9..

10

.7

41

"

20

04

00

10

00

B^(

Gau

ss)

Fig.

31V

aria

tion

ofR

eson

ant

Fre

quen

cyof

and

Res

onan

ces

with

Mag

netic

Fiel

d,fp

=2.

4G

c.

because it was masked by the ^ stronger resonance. Above

1050 gauss it was again not possible to measure it because here it was

masked by the resonance.

C. NOISE AND DRIFT

Although noise fluctuations of the parameters of the mercury-

vapor discharge positive colvimn did not limit the measurements reported

in this report, they are of interest to the development engineer who may

be contemplating the use of this discharge in some application. Slow-time

drift of the parameters, probably due to small changes in the vapor pres

sure, were troublesome and constituted the chief source of error in the

propagation constant measurements. Both of these detractors are discussed

below.

For a mode propagating in the plasma waveguide, the total power P

that flows across a cross-section of the system may be represented as the

sum of two terms,

P = P + P (5. 4)s n

where P is the power at the transmitter or signal frequency, and P^ is

the unwanted power at all other frequencies which we call the noise power.

We have found experimentally that the noise power output of the transmitter

(signal source) is very small compared to P^, so that P^ is nearly allgenerated in the plasma waveguide. If we probe the mode with the detection

probe, we shall assume that the signal power and the noise power absorbed

by the probe and which travel down the coaxial transmission line to the mea

suring apparatus, will have the same ratio as P^ and P^ at the positionof the probe.

It has been possible to measure 10 log n = 0 and

n = 1 modes described in V. A. Here, we use the microwave bridge shown

-84-

in Fig. 14, where the detector was a high-sensitivity receiver capable

of accurate measurement of relative power levels. Disconnecting the

reference bridge arm which contains the calibrated line-stretcher and

attenuator, the power level of the signal arriving from the detection probe

is noted. This power is proportional to the total power P = + P^. Wenow connect the reference arm of the bridge and balance out the transmitter

frequency component of the signal. Thus, the only power which can now

reach the receiver is proportional to P^ so that we are able to measure

10 log (P/P^). From this, 10 log easily obtained.Measurements of this type have been made on both the n = 0 and

n = 1 modes. We find that 10 log independent of the input power

level to the plasma waveguide. Here, the input power was varied from about

1 milliwatt to . 1 milliwatt.

In Fig. 32 we show the variation of 10 log ^ function of

frequency for the detection probe far from the double-ring coupler. The

noise power increases relative to the signal power for the n = 0 mode as

0)^^ is approached. For the n =1mode, 10 log nearly constantacross its band. In Fig. 33, 10 log shown as a function of probe

position at discreet frequencies. In all cases, the noise power decreases

relative to the signal power as the input coupler is approached.

We have examined the signal emerging from the detection probe with

a spectr\im analyzer which has a 1000 cps sweep rate. This means that

the spectrum analyzer will display the spectrum of the signal on the cathode-

ray tube screen at millisecond intervals. From this it was possible to

determine that if is the radial field component which excites the detec

tion probe at time t^, then a millisecond later at t^, the radial field component will be slightly different in amplitude and phase from A

millisecond later at t^, slightly different fromE E ... are shown in Fig. 34. Thus, there appears to be a small

rZ r3 °

-85-

I

00ON1

Tube No. 3

fps2.4(Gc)

(Iq'.SO amp)BQ-3.5gaussoxiol position sQ

0.9 1.0

FREQUENCY f (Gc)Fig. 32 Signal to Noise Ratio for n = 0,1 Modes vs.

Frequency; f = 2. 4 Gc.IT

I 00

-J 1

16

f=

.8

50

Gc

f=

.98

0G

c

nee

Alo

ng

Ax

is(i

n.)

Tu

be

No

.

fp=2

.4G

o.B

=3

.5

Gau

ss

(1=

.6

0am

p.)

Fig.

33Si

gnal

toN

oise

Rat

iovs

.D

ista

nce

Alo

ngPl

asm

aW

aveg

uide

.Th

ePo

sitio

nIn

dic

ated

by0

isN

ear

the

Ter

min

atio

n.

Fig. 34 Phaser Diagram of Ej. at Five Observation Times.

phase and amplitude modulation of the signal, and therefore, is the

power in the side-bands which exist by virtue of this modulation. It was

also possible to determine from observations with the spectrum analyzer-

that the amplitude and phase modulation frequencies were less than 1 Mc.

We now propose a simple model of the plasma which will explain all

the observations described above. Assume that the electron number density

n consists of a part which is independent of time, ^ small time-

varying part, A so that

n = n + A n (t)e eo e

(5.5)

n may be regarded as a time-average of n while An (t) is a noiseeo

fluctuation of n whose frequency components are less than 1 Mc. Assumee

that A n (t) is independent of position so that the electron density every-e

where in the positive column is fluctuating coherently.

-88-

Equation (5.5) for n will lead to a plasma angular frequencye

0) =0) + A CO (t) (5'6)p po p

and a cut-off angular frequency

(0 —(co ) + Aco (t) . (5*7)CO CO O CO

If the n = 0 mode or the n = 1 mode is propagating in the system with a

propagation constant p and an attenuation constant a, then

|3 =|3^ +AP(t) , (5. 8)and

or = Of + A Qr(t) . (5* 9)o

In the above expressions a quantity with a subscript "o" is a time

average quantity while a quantity preceded by A is a noise fluctua

tion of the quantity due to A

If the probe is a distance L from the coupler, then E^, thecomponent of electric field to which the probe is sensitive, will be

related to E at the coupler by the factorr

exp(-QrI-» - ipL) =exp (-or^L - ip^L) exp [-Aar(t)L] exp [-iAp(t)Ij] •(5.10)

The factors

exp [-Aa(t)L ] and exp [-iAp(t)L]

will account for the amplitude and phase modulation, respectively, of

the wave and P will be the resulting power in the side-bands,n

-89-

It is clear now that 10 log P /P will be independent of thes n

input power supplied by the transmitter because the system is linear.

To explain qualitatively the variation of 10 log P^

Figs. 32 and 33, consider a particular angular frequency in the

noise frequency spectrum. Suppose that the component of An^(t) atthis frequency is An^^^ sin co^t, so that the factors producing amplitude and phase modulation at in Eq. (5.10) are,

exp [ -Aa L sin (cat+ 0')] and exp [ -iAp, L sin (ut + 0")]k ^

respectively. The greater the more power there

will be in the side-bands associated with and the greater will be the

contribution to P at co, . The same is true at all other noise frequen-n k

cies.

Referring to the Brillouin diagrams for the n = 0 and n = 1 modes

shown in Fig. 28, it can be seen that a fluctuation of at the noise

frequency of amplitude A will cause both and Aor^^ toincrease as the transmitter frequency approaches Thus, for

constant P and L, P will increase as the transmitter frequency ap-s n

proaches This is in agreement with the variation of 10 log P /P^shown in Fig. 32. At a constant transmitter frequency, if the detection

probe is moved toward the coupler so that L decreases, then Aofj^L andA 3 L will decrease and therefore P„ will decrease, which is in agree-

r-k n

ment with the variation of 10 log P /P shown in Fig. 33.s n

The model of the plasma which we have made use of above, arid

which is summarized by Eq. (5. 5), has been pr.edicted approximately34

by Crawford and Lawson. They find that in a niercury-vapor discharge,

except for a region extending 8 in. beyond the cathode, there is a noise

fluctuation of the electron number density which is coherent from point to

-90-

point. The diameter of their discharge was 2 cm. and they operated

at a discharge current of approximately . 1 amp. All measurements

reported in this section were made near the anode, where presumably

the fluctuations of n were coherent.e

Crawford and Lawson postulate that there is a source of noise

voltages in the region near the cathode and that waves of noise voltage

propagate down the positive column with phase velocities greater than

10*^ cm/sec. The spectrum of the noise voltage frequencies shows predominate components at 60 Kc. and 105 Kc. at . 1 amp. discharge current.

They have measured relative fluctuations in n^ and find that they becomesomewhat greater as the cathode is approached.

We conclude the discussion of noise by noting that for measure

ments in which a d. c. magnetic field was present, for example, in

measurements on the TE. ,+ and TE - modes, 10 log P »11 11 is XI.

the noise power is very large and implies that the presence of a magnetic

field introduces a new source of noise into the system.

Slow drift of the phase shift of a mode from the coupler to the de

tection probe was the chief cause of error in the propagation constant

measurements. This is probably due to a drift in the vapor pressure

which causes the electron number density n^ to vary. We have beenable to estimate that drift in phase shift causes a + 3 percent error in

the measurement of (3. The amplitude of a mode at the position of the

detection probe was quite constant, varying less than . 1 db over a 7 min

ute observation time.

-91-

VI. SUMMARY AND CONCLUSIONS

In this report we have examined experimentally the properties

of certain modes in a plasma waveguide of circular symmetry. When

the magnetic field is essentially zero, a backward-wave pass-band

mode has been found which is a n = 1 surface wave. Using the expres

sion for the permittivity tensor which neglects the effects of thermal

motion of the electrons and ions of the plasma and also collisions be

tween plasma components, it has been possible to predict the propaga

tion constant p of the n = 1 mode and also the low-pass n = 0 mode

with considerable precision.

This result was predicted in 11. There it was indicated

that the electron temperatures typically found in a mercury-vapor dis

charge positive column should have little effect on the values of the

elements of the permittivity tensor. In that section we further pointed

out that for the n = 0 mode, the collision frequency v has negligible

effect on the propagation constant p except where the attenuation con

stant is very large. Specifically, for the n = 0 mode propagating in a

two region system, if

2as < I (2. 37)

the neglect of collisions and of the thermal velocities of the plasma

components will not lead to appreciable error in the calculation of the

propagation constant. We would not be surprised to find that the same

criterion could be applied to a three-region system and to other modes

besides the n = 0. The measurements reported above are experimental

evidence that this criterion is applicable to the n = 0 and n = 1 modes

in a three-region system.

-92-

Both the n = 0 and n = 1 modes were excited by a novel coupler,

called a double-ring coupler , which is geometrically symmetric. It

was presumed that the non-symmetric n = 1 mode was excited by vir

tue of the electric field between its two rings being non - symmetric;

i. e. , the field varies as a function of the azimuthal coordinate. At

the point where the two rings are attached to the exciting coaxial trans

mission line, the electric field probably is greatest and is a minimum

on the opposite side of the rings. That is, if <j), the azimuthal coor

dinate, is zero at the position where the rings are excited, then the

field between the rings is a maximum at <)> = 0 and a minimum at

<|> =180°. It is suggested that if it is desired to make the electric fieldindependent of <j) so that it will not excite the non-symmetric n ~1 mode^

then the rings could be squeezed together at <|) = 180 so as to boost the

electric field there. A better way of saying this is that instead of the

planes of the two rings being parallel, they should be set at an angle

such that the rings will be closer together at ^ = 180 than at <)) = 0.

Conversely, the field may be made more non-iniform so that the n —1

mode will be excited more effectively by setting the planes of the rings

at such an angle that the rings are farther apart at <j) =180° than at(|) = 0.

Noise measurements have been made on both the n = 0 and the n = 1

modes. We have found that the signal-to-noise ratio, 10 log Pg'^^n'

better than 30 db for the n = 0 mode if we are in the region o-f constant

phase velocity. It will drop to about 12 db as the cut-off angular frequency

o) is approached. For the n = 1 mode, 10 log P /P is low, of the orderCO ® ^

of 7 db. This ratio is independent of input transmitter power. The results

of the noise measurements can be explained by a simple model of the plas

ma, and independent verification of the validity of this model may be found35

in the work of Crawford and Lawson.

-93-

For the case of an anisotropic plasma, the propagation constants

for the and the modes have been found and the results

are in qualitative agreement with the theory. These measurements

were made in a resonant system. It was found that each waveguide

mode would give rise to a series of resonances, and

These resonances were easily excited by a simple probe, like the de

tection probe, which coupled to E^ of a mode.

It has been pointed out in Section II that the glass tube which con

fines the plasma has a significant effect on the propagation constants ofboth the n = 0 and the n = 1 modes. In fact, it is due to the presence of

the glass tube that the n =1 mode is a backward-wave mode. We mightexpect, then, that by the proper choice of the ratios of a/b and a/c, and€ (see Fig. 1), we can make the n = 1 mode not only a backward-wave

as was the case investigated in this report, but also a forward-wave as

Trivelpiece shows, ^ and finally a forward-wave over part of the rangeof p and a backward-wave over the rest of the range of p. The lattervariation of p would be accomplished by using a rather high permitti

vity €_ and making the wall of the tube very thin.

It is not difficult to conceive of practical devices based on the

modes that have been investigated in this paper. For example, it is

known that the interaction of an electron beam with the n = 0 mode will35 , .

result in forward-wave amplification of an applied signal, while it

is possible that interaction of an electron beam with the n =1 mode might35

result in backwjard-wave oscillation (Boyd'"^"shows that the beam can be

focused without the use of a magnetic field). Various passive devices

are also possible such as a phase shifter or switch.

The chief difficulty in constructing these devices is the unavail

ability of a suitable plasma medium. It should be clear that the mercury-vapor discharge is unsuitable because of three detracting characteristics.

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drift of parameters, significant noise modulation of the signal, end loss,particularly for the n =1mode. It is conceivable that the first two couldbe reduced, but it is doubtful if they could be entirely eliminated. For

example, drift could be reduced by careful temperature control of the34

discharge tube. Since noise voltages evidentally propagate as waves,

it might be possible to interrupt this propagation by placing a reflectingscreen across the discharge. It was found early in this project that a

course floating screen of four fine tungsten wires placed across the discharge did not appear to alter its chara.cteristics. No noise measurements were made at that time.

To the best knowledge of the author, the most superior plasma forpractical devices that exists at the present time is a cesium thermalplasma constructed by G- S. Kino and his co-workers at Stanford University. ^ Through temperature control of the enclosing glass envelope,drift has been effectively eliminated. By ensuring that an electron sheathexists around the plasma-producing tungsten buttons or spirals, and takingcertain other precautions, a plasma has been created which is oscillationfree. Because the percent ionization of the cesium is nearly 100 percentand because the temperature of the plasma components is about 1000°C,the collision frequency v is of the order of a few megacycles.

This plasma also has its detracting aspects. The equipment neededto provide temperature control is ciimbersome. Also, the only way thatthe electron density can be altered is to change the temperature of thecesium vapor, the time necessary to accomplish this being of the orderof minutes. Also, a high magnetic field is needed to confine the plasma.From a manufacturer's point of view, this is not appealing.

Thus, the chief problem in device work at the present time-- theproblem that stands in the way of the modes described here and elsewhere

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being put to practical use--is the unavailability of a good plasma. Until

this is developed, traveling-wave tubes will always be more appealing

than the plasma amplifier and ferrite phase shifters will be preferable

to plasma phase shifters.

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VII. APPENDIX - ELECTRICAL CHARACTERISTICS OF A LOW

PRESSURE DISCHARGE IN THE ABSENCE OF A

MAGNETIC FIELD

In a hot-cathode arc discharge which is contained in a long dielec

tric tube, nearly all of the volume of the discharge has characteristics

which allow it to be classified as the positive column. Near the cathode,

anode, and dielectric tube, however, there are boundary regions with

properties quite unlike the positive column. The characteristics of these

regions have been described in the literature and have recently been sum-36

marized by Crawford and Kino.

Directly in front of the cathode is a narrow region of less than 1 mm.

thickness. The portion of this region nearest the cathode is an electron

sheath. The potential drop across this region is about that of the ioniza-

tion potential (10. 4 volts for mercury). The motion of the electrons in

this region is similar to that in a diode with the result that the electrons

emerge from this region with highly directed velocities, very little

velocity spread, and energies of about the ionization potential. These

electrons are known as primary electrons.

Next, there is a region where so-called ultimate electrons exist

which have approximately an isotropic Maxwellian velocity distribution.

These are much more numerous than the primary electrons emerging

from the "diode" region. Beyond this region the primary electrons become

scattered and transformed into ultimate electrons. The mechanism of

scattering appears to be associated with a high frequency plasma oscilla-37

tion. Workers have reported observing a bright meniscus-shaped area

in this region, although it was not apparent in the discharge tubes used in

the experiments reported here. Beyond this region only ultimate elec

trons exist and this is the region that occupies almost all of the discharge

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volume, the positive column.

Near the anode there is a sheath, again of the order of a milli

meter in length, the polarity of which depends on the thermal current

density in the positive column, the size of the anode, and the total29

current that is being passed through the tube.

Finally, along the dielectric tube, a narrow, negative sheath

exists which is due to equal electron and ion current flow to the wall.

This will be discussed in more detail below.

The properties of the positive column do not depend on its length.

We assume that it is homogeneous except near its boundaries. The

concentration of electrons is nearly equal to that of ions so that the

positive column is essentially electrically neutral and fits our defini

tion of a plasma.

The average temperature of the electrons in the positive column

is of the order of 20, GOG^K. This temperature is essentially indepen-29

dent of radial position of the electon. The temperature of the ions is

slightly greater than the wall temperature, while the temperature of the

neutral atoms is that of the wall. The degree of ionization of the mer

cury vapor is always small (of the order of a few tenths of a percent

for mercury vapor). Consequently, because the thermal velocity of the

electrons is much larger than that of the ions, the primary collisional

process is either electron-neutral collisions or electron-wall collisions.

These will be typically of the order of tens or hundreds of megacycles.

One of the interesting features of the electron behavior is that their

velocity distribution is approximately Maxwellian, as can be demonstrated29

by probe measurements. Druyvesteyn and Penning suggest that high

frequency oscillations in the plasma may be essential for the establish

ment of a Maxwellian distribution.

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A Maxwellian velocity distribution is in a sense paradoxical

("Langmuir's Paradox") because at low pressures electrons in the

high-velocity tail of the Maxwellian curve should escape through

the sheath at the wall to recombine with ions at the wall. High-fre

quency oscillations in the sheath have been postulated as the mechan-38

ism which prevents this from happening.

Recombination of ions and electrons occurs primarily at the

dielectric wall. It is unlikely that recombination will occur in the

body of the discharge because the laws of energy and momentum con

servation cannot be simultaneously met in a simple two-body recom-

bining collision. Thus su three-body collision or a two-body collision39

with the emission of a photon is necessary for recombination, both

occurrences being improbable in the discharge body. The wall, how

ever, constitutes a convenient third body.

Equilibrium requires that for each electron and ion lost through

recombination another pair must be created. This occurs in the body

of the positive column through ionization of neutral atoms by an axialelectric field. This field also causes electrons and ions to have drift

velocities which are superimposed on their thermal velocities.

The production of electron-ion pairs in the body of the positive

column and the loss of electron-ion pairs at the wall implies that a

current of both electrons and ions must flow radially outward to the

wall. In order to regulate the flow to just the right amount, a radial

electric field exists in the direction of the wall. This tends to speed

up the low-velocity ions and slow down the high-velocity electrons so

that the radial current densities of these two particles are exactly

equal and of just the right amount to supply the loss through recom

bination. This radial electric field originates on net positive charge

in the body of the positive coliimn, implying that the positive column

is not precisely electrically neutral, but is slightly positive. In fact,

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the positive column is neutral at the center and becomes increasingly

positive toward the wall. Since there is no field outside the dielectric

discharge tube, we must postulate a layer of negative charge (electrons)

on the wall on which the electric field lines terminate. This layer of

charge occurs, in fact, when the tube is first turned on, at which time

electrons quickly diffuse to the wall.

Note that the potential drop, in going from the relatively neutral

body of the positive column to the wall, is negative. According to the

Boltzmann distribution, a potential which decreases toward the wall re

quires that the electron density must also decrease toward the wall.

Indeed, Killian^^ has found using probe techniques that the electron density in a mercury-vapor discharge at the wall will have decreased of the

order of 20 percent from its value on the axis.

The effect of increasing the discharge current chiefly increases

the charge density only and has little effect on the electric field in the

positive column or on the temperature of the charged particles. Thus

the drift velocity is also independent of current. This means that the

charge density is proportional to the discharge current. Since the plasma

frequency is proportional to the square-root of the charge density, the

plasma frequency should be proportional to the square-root of the discharge

current.

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